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101. According to Newton’s second law for a system of particles, the motion of the center of mass is governed by:
ⓐ. \(M \vec{a}_{cm} = \sum \vec{F}_{int}\)
ⓑ. \(M \vec{a}_{cm} = \sum \vec{F}_{ext}\)
ⓒ. \(M \vec{a}_{cm} = 0\)
ⓓ. \(M \vec{a}_{cm} = \vec{v}_{cm}\)
Correct Answer: \(M \vec{a}_{cm} = \sum \vec{F}_{ext}\)
Explanation: Internal forces cancel in pairs due to Newton’s third law, so only external forces contribute to the acceleration of the center of mass. Thus, the system behaves like a particle of mass \(M\) placed at the COM.
102. If no external force acts on a system of particles, the velocity of the center of mass will:
ⓐ. Continuously increase
ⓑ. Continuously decrease
ⓒ. Remain constant
ⓓ. Oscillate periodically
Correct Answer: Remain constant
Explanation: With no external force, \(M \vec{a}_{cm} = 0\). Hence, \(\vec{v}_{cm}\) is constant, meaning the COM moves in a straight line with uniform velocity or stays at rest.
103. For a two-particle system of masses \(m_1\) and \(m_2\) with velocities \(\vec{v}_1\) and \(\vec{v}_2\), the velocity of the center of mass is:
ⓐ. \(\frac{v_1 + v_2}{2}\)
ⓑ. \(\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2}\)
ⓒ. \(\frac{m_1 + m_2}{v_1 + v_2}\)
ⓓ. \(m_1 v_1 – m_2 v_2\)
Correct Answer: \(\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2}\)
Explanation: The velocity of the center of mass is the total momentum divided by the total mass. This formula shows how both mass and velocity of particles affect COM motion.
104. The momentum of a system of particles can be expressed as:
ⓐ. \(\vec{P} = M \vec{v}_{cm}\)
ⓑ. \(\vec{P} = I \omega\)
ⓒ. \(\vec{P} = \sum F_{ext}\)
ⓓ. \(\vec{P} = \frac{1}{2} M v^2\)
Correct Answer: \(\vec{P} = M \vec{v}_{cm}\)
Explanation: The total linear momentum of the system equals the product of total mass and the velocity of its center of mass. This reduces the system’s motion to that of a single equivalent particle.
105. If a firecracker explodes in space into multiple fragments, what happens to the velocity of the center of mass?
ⓐ. It decreases to zero
ⓑ. It increases continuously
ⓒ. It remains unchanged
ⓓ. It oscillates with fragments
Correct Answer: It remains unchanged
Explanation: In the absence of external forces, the COM continues with the same velocity as before the explosion. Internal forces (explosion forces) do not affect the COM motion.
106. Which of the following is correct regarding kinetic energy of a system of particles?
ⓐ. It is always equal to the kinetic energy of the center of mass only
ⓑ. It is the sum of kinetic energy of the COM motion and the kinetic energy relative to the COM
ⓒ. It depends only on external forces
ⓓ. It depends only on mass of the COM
Correct Answer: It is the sum of kinetic energy of the COM motion and the kinetic energy relative to the COM
Explanation: Total kinetic energy \(K = \frac{1}{2} M v_{cm}^2 + \sum \frac{1}{2} m_i v_i’^2\), where \(v_i’\) are velocities relative to the COM. This decomposition separates overall system motion and internal motion.
107. For a system of particles, the acceleration of the center of mass depends on:
ⓐ. The net internal force only
ⓑ. The external forces acting on the system
ⓒ. The position of the heaviest particle
ⓓ. The total number of particles
Correct Answer: The external forces acting on the system
Explanation: Since internal forces cancel, only the vector sum of external forces decides the acceleration of the COM. Thus, regardless of internal interactions, the COM follows the effect of external forces.
108. If two ice skaters push off each other on frictionless ice, the center of mass of the system will:
ⓐ. Move toward the heavier skater
ⓑ. Move toward the lighter skater
ⓒ. Stay fixed in position
ⓓ. Move in circles
Correct Answer: Stay fixed in position
Explanation: Internal forces (push) act in equal and opposite directions. With no external force, the COM remains stationary even though both skaters move in opposite directions.
109. Which equation expresses the relation between external impulse and change in COM momentum?
ⓐ. \(J = \Delta (M v_{cm})\)
ⓑ. \(J = M v_{cm}\)
ⓒ. \(J = I \omega\)
ⓓ. \(J = \tau \Delta t\)
Correct Answer: \(J = \Delta (M v_{cm})\)
Explanation: The impulse–momentum theorem states that the external impulse \(J = \int F_{ext} dt\) equals the change in total momentum of the system, or equivalently, the change in COM momentum.
110. Which of the following statements is true about the equations of motion for the center of mass?
ⓐ. They are different from Newton’s laws
ⓑ. They are identical to Newton’s laws applied to a single particle of mass \(M\) located at the COM
ⓒ. They apply only to rigid bodies
ⓓ. They ignore external forces completely
Correct Answer: They are identical to Newton’s laws applied to a single particle of mass \(M\) located at the COM
Explanation: The motion of a system’s COM can be analyzed as if all mass were concentrated at the COM and subjected to the net external force. This equivalence greatly simplifies analysis.
111. In a uniform gravitational field, the acceleration of the center of mass of a system of particles is:
ⓐ. Zero always
ⓑ. Equal to the acceleration due to gravity \(g\)
ⓒ. Equal to the average acceleration of all particles
ⓓ. Dependent only on the heaviest particle
Correct Answer: Equal to the acceleration due to gravity \(g\)
Explanation: Each particle experiences the same acceleration \(g\) in a uniform field. Thus, the center of mass also accelerates downward with acceleration \(g\), independent of mass distribution.
112. If two bodies of unequal masses are connected and fall freely in a uniform gravitational field, the center of mass of the system will:
ⓐ. Fall faster toward the heavier body
ⓑ. Fall slower toward the lighter body
ⓒ. Fall with the same acceleration \(g\) as any free body
ⓓ. Remain stationary
Correct Answer: Fall with the same acceleration \(g\) as any free body
Explanation: Regardless of the individual masses, the system’s COM behaves like a single particle of total mass \(M\) acted upon by gravity, so it falls with acceleration \(g\).
113. For a projectile launched into a uniform gravitational field, the path of its center of mass is:
ⓐ. A straight vertical line
ⓑ. A straight horizontal line
ⓒ. A parabola
ⓓ. A circle
Correct Answer: A parabola
Explanation: The COM follows the same laws of motion as a single particle under uniform gravity. Thus, its path is parabolic, identical to that of a single projectile.
114. If a bomb projected in air explodes into fragments, the path of the center of mass of all fragments will:
ⓐ. Deviate upward
ⓑ. Deviate sideways
ⓒ. Continue along the same parabolic trajectory as if no explosion occurred
ⓓ. Fall vertically down
Correct Answer: Continue along the same parabolic trajectory as if no explosion occurred
Explanation: Internal explosion forces cancel. The COM of the fragments continues in the parabolic path dictated only by the external gravitational force.
115. In a uniform gravitational field, the potential energy of a system of particles is:
ⓐ. The average of the potential energies of all particles
ⓑ. The sum of the potential energies of individual particles
ⓒ. The potential energy of the total mass concentrated at the center of mass
ⓓ. Zero always
Correct Answer: The potential energy of the total mass concentrated at the center of mass
Explanation: Total potential energy = \(M g y_{cm}\), as if all the mass were concentrated at the COM at height \(y_{cm}\). This simplifies calculations in gravitational fields.
116. If a rigid body is suspended freely in a uniform gravitational field, it will come to rest in which position?
ⓐ. With its COM at the highest possible point
ⓑ. With its COM directly below the point of suspension
ⓒ. With its COM directly above the suspension point
ⓓ. At any random orientation
Correct Answer: With its COM directly below the point of suspension
Explanation: For stable equilibrium, the line of action of weight (through the COM) must pass through the suspension point. Hence, the COM settles directly below it.
117. When a system of particles is placed in a uniform gravitational field, the motion of the COM is equivalent to:
ⓐ. A single particle of mass \(M\) placed at COM acted upon by gravity
ⓑ. A single particle of the heaviest mass only
ⓒ. Independent motion of each particle
ⓓ. Zero motion always
Correct Answer: A single particle of mass \(M\) placed at COM acted upon by gravity
Explanation: In uniform gravity, the COM behaves as if the entire mass is concentrated at that point. Thus, its motion is equivalent to a particle of mass \(M\) at the COM.
118. If three equal masses are placed at the vertices of a triangle in a uniform gravitational field, the total gravitational force on the system is:
ⓐ. Zero
ⓑ. \(3mg\) acting at the centroid of the triangle
ⓒ. \(mg\) acting at one vertex
ⓓ. \(2mg\) acting at the midpoint of one side
Correct Answer: \(3mg\) acting at the centroid of the triangle
Explanation: Each mass experiences a force \(mg\). The resultant is \(3mg\), and by symmetry it acts through the centroid, which is the COM of the system.
119. When analyzing the equilibrium of bodies in a uniform gravitational field, which point is most convenient for calculations?
ⓐ. Any vertex of the body
ⓑ. The geometric center
ⓒ. The center of mass
ⓓ. A point at infinity
Correct Answer: The center of mass
Explanation: Since the entire weight of the body can be considered to act through the COM in uniform gravity, analyzing equilibrium with respect to the COM is simplest and most accurate.
120. Why does a freely falling body in uniform gravity appear weightless to its internal parts?
ⓐ. Because gravity is absent inside the body
ⓑ. Because all parts accelerate equally with gravity, so relative force between them vanishes
ⓒ. Because the COM remains fixed in space
ⓓ. Because the heaviest part cancels the lighter parts
Correct Answer: Because all parts accelerate equally with gravity, so relative force between them vanishes
Explanation: In free fall, every part of the system experiences the same acceleration \(g\). Since no internal stresses are needed to maintain this, the body appears weightless to its constituents.
121. In a collision between two particles, what determines the motion of the center of mass of the system?
ⓐ. Internal impulsive forces during the collision
ⓑ. External forces acting on the system
ⓒ. The kinetic energy of the system
ⓓ. The heavier particle only
Correct Answer: External forces acting on the system
Explanation: Internal forces obey Newton’s third law and cancel in pairs. Therefore, only external forces affect the motion of the center of mass. During a collision, the COM continues its motion unaffected by internal impulsive forces.
122. If two bodies collide on a frictionless horizontal surface, the velocity of their center of mass after the collision will:
ⓐ. Increase
ⓑ. Decrease
ⓒ. Remain constant
ⓓ. Become zero
Correct Answer: Remain constant
Explanation: With no external horizontal force, momentum of the system is conserved. Hence, the velocity of the COM remains unchanged, regardless of the type of collision (elastic or inelastic).
123. When two objects collide and stick together (perfectly inelastic collision), the center of mass of the system:
ⓐ. Moves randomly
ⓑ. Comes to rest
ⓒ. Moves with constant velocity as before collision
ⓓ. Accelerates indefinitely
Correct Answer: Moves with constant velocity as before collision
Explanation: Even in an inelastic collision, internal forces cannot affect the motion of the COM. Its velocity remains the same as just before the collision, provided no external force acts.
124. In elastic collisions, how does the motion of the center of mass compare before and after the collision?
ⓐ. It changes direction
ⓑ. It accelerates due to internal forces
ⓒ. It remains the same
ⓓ. It comes to rest
Correct Answer: It remains the same
Explanation: Elastic collisions conserve both momentum and kinetic energy, but neither affects COM motion. With no external force, the velocity of the COM remains unchanged.
125. Two bodies of masses \(m_1\) and \(m_2\) moving toward each other collide elastically. What happens to the velocity of the center of mass?
ⓐ. It increases
ⓑ. It decreases
ⓒ. It remains unchanged
ⓓ. It becomes equal to the velocity of the heavier body
Correct Answer: It remains unchanged
Explanation: In the absence of external force, the velocity of the COM remains unaffected by the details of the collision. Internal interaction only redistributes velocities relative to the COM.
126. In a system of colliding particles, why is the center of mass frame often used for analysis?
ⓐ. Because it reduces mass of the particles
ⓑ. Because total momentum is zero in the COM frame
ⓒ. Because external forces vanish in this frame
ⓓ. Because velocities become infinite in this frame
Correct Answer: Because total momentum is zero in the COM frame
Explanation: In the COM frame, the vector sum of particle momenta is zero. This simplifies analysis of collisions since velocities are symmetric and relative motion is easier to calculate.
127. If two equal masses collide elastically in one dimension, the center of mass:
ⓐ. Comes to rest after the collision
ⓑ. Moves with constant velocity unaffected by collision
ⓒ. Reverses its velocity
ⓓ. Accelerates continuously
Correct Answer: Moves with constant velocity unaffected by collision
Explanation: Since both masses are equal and forces are internal, the COM velocity is unchanged by the collision. The masses may exchange velocities, but the COM keeps moving as before.
128. In an explosion (treated as a reverse collision), the fragments fly apart in different directions. The center of mass of the system:
ⓐ. Moves unpredictably
ⓑ. Remains stationary if initial momentum was zero
ⓒ. Moves toward the heaviest fragment
ⓓ. Accelerates continuously
Correct Answer: Remains stationary if initial momentum was zero
Explanation: In an explosion without external forces, the system’s COM remains at rest. The fragments move in such a way that their momenta cancel out, keeping the COM fixed.
129. If two bodies collide on a rough horizontal surface with friction acting, the velocity of the COM:
ⓐ. Changes due to the external force of friction
ⓑ. Remains unchanged
ⓒ. Is independent of external forces
ⓓ. Becomes zero always
Correct Answer: Changes due to the external force of friction
Explanation: Friction is an external force on the system, so it alters the net momentum. Consequently, the velocity of the COM changes during the collision.
130. In a head-on elastic collision between two masses, the kinetic energy of the COM motion:
ⓐ. Is always conserved
ⓑ. Is always lost
ⓒ. Is transformed into internal energy
ⓓ. Is zero
Correct Answer: Is always conserved
Explanation: The kinetic energy of the COM motion depends only on external forces. Since internal forces in a collision do not affect the COM, the kinetic energy of the COM remains conserved, even if some internal kinetic energy is lost in inelastic collisions.
131. Why is the concept of center of mass important in space missions?
ⓐ. It decides the color of the spacecraft
ⓑ. It determines the trajectory of the spacecraft in orbit
ⓒ. It reduces the spacecraft’s total weight
ⓓ. It increases fuel efficiency automatically
Correct Answer: It determines the trajectory of the spacecraft in orbit
Explanation: In space, external forces such as gravity act on the spacecraft as if its entire mass were concentrated at the center of mass. Thus, the path of the spacecraft depends on the motion of its COM.
132. Why do satellites carry gyroscopes and reaction wheels?
ⓐ. To increase their velocity
ⓑ. To control orientation by conserving angular momentum
ⓒ. To reduce their total weight
ⓓ. To make them spin randomly
Correct Answer: To control orientation by conserving angular momentum
Explanation: Gyroscopes and reaction wheels exploit the conservation of angular momentum. By changing spin rates of internal wheels, satellites adjust their orientation without expelling fuel, using COM and rotational dynamics.
133. If a spacecraft ejects gas in one direction, what happens to the spacecraft’s center of mass motion?
ⓐ. The COM shifts backward
ⓑ. The COM remains unaffected, but the spacecraft gains velocity in the opposite direction
ⓒ. The COM stops moving
ⓓ. The COM moves with infinite speed
Correct Answer: The COM remains unaffected, but the spacecraft gains velocity in the opposite direction
Explanation: The expelled gas and spacecraft form a system. The COM continues with uniform motion (if no external force acts), but relative to the COM, the spacecraft moves opposite to the exhaust direction.
134. Why must fuel tanks in rockets be aligned with the rocket’s center of mass?
ⓐ. To reduce gravitational force
ⓑ. To maintain stability and avoid unwanted torques
ⓒ. To increase velocity directly
ⓓ. To decrease fuel consumption
Correct Answer: To maintain stability and avoid unwanted torques
Explanation: If fuel is stored away from the COM, uneven consumption shifts the COM and creates torque, destabilizing the rocket. Aligning tanks along the COM axis ensures smooth and stable flight.
135. Which principle of rotational motion explains why spinning satellites remain stable in space?
ⓐ. Newton’s first law
ⓑ. Conservation of angular momentum
ⓒ. Law of gravitation
ⓓ. Hooke’s law
Correct Answer: Conservation of angular momentum
Explanation: A spinning satellite resists changes in its axis of rotation due to conservation of angular momentum. This principle helps maintain stability in space missions without continuous fuel usage.
136. In robotics, why is knowledge of the center of mass crucial?
ⓐ. It reduces cost of construction
ⓑ. It ensures balance and stability during walking or movement
ⓒ. It determines speed of computation
ⓓ. It increases battery life
Correct Answer: It ensures balance and stability during walking or movement
Explanation: Robots must keep their COM within the base of support to avoid falling. Engineers design walking algorithms that continuously adjust robot posture to keep the COM stable.
137. Why do astronauts in space stations feel weightless even though gravity acts on them?
ⓐ. Gravity is zero in space
ⓑ. They are outside Earth’s gravitational influence
ⓒ. Both astronauts and the station fall together, so their COM follows free fall in orbit
ⓓ. Their mass becomes zero
Correct Answer: Both astronauts and the station fall together, so their COM follows free fall in orbit
Explanation: In orbit, the station and astronauts continuously fall toward Earth with the same acceleration. Since their COM follows free fall, they feel weightless relative to each other.
138. Which application of COM is most important in designing humanoid robots?
ⓐ. Center of gravity must lie above the head
ⓑ. Center of mass must stay within the base of support during motion
ⓒ. COM should remain fixed at the waist
ⓓ. COM must always remain outside the body
Correct Answer: Center of mass must stay within the base of support during motion
Explanation: For stability, humanoid robots must shift their COM dynamically during walking, running, or lifting. If the COM moves outside the base of support, the robot topples.
139. In rocket propulsion, what ensures conservation of momentum of the system?
ⓐ. External gravitational pull
ⓑ. Expulsion of fuel gases backward and forward motion of rocket
ⓒ. Increase in mass of the rocket
ⓓ. Rotation of the rocket
Correct Answer: Expulsion of fuel gases backward and forward motion of rocket
Explanation: The rocket and exhaust gases form an isolated system. The COM of the system moves uniformly, while conservation of momentum ensures that expelling fuel backward propels the rocket forward.
140. How is COM used in controlling robotic arms in manufacturing?
ⓐ. By ignoring mass distribution
ⓑ. By programming the arm so its COM remains balanced as it moves
ⓒ. By fixing COM at the shoulder joint only
ⓓ. By allowing random shifts in COM
Correct Answer: By programming the arm so its COM remains balanced as it moves
Explanation: Robotic arms must maintain balance to avoid vibrations or instability. By keeping track of COM during movements, engineers ensure smooth operation and prevent tipping or mechanical stress.
141. Which of the following best defines linear momentum of a particle?
ⓐ. Product of mass and acceleration
ⓑ. Product of mass and velocity
ⓒ. Product of force and time
ⓓ. Product of displacement and mass
Correct Answer: Product of mass and velocity
Explanation: Linear momentum \(\vec{p}\) is defined as \(\vec{p} = m \vec{v}\), where \(m\) is the mass and \(\vec{v}\) is the velocity of the particle. It is a vector quantity pointing in the direction of velocity.
142. The SI unit of linear momentum is:
ⓐ. Newton
ⓑ. Joule
ⓒ. \(\text{kg·m/s}\)
ⓓ. \(\text{N·m}\)
Correct Answer: \(\text{kg·m/s}\)
Explanation: Since linear momentum is mass \((kg)\) multiplied by velocity \((m/s)\), its unit is \(kg·m/s\). It can also be expressed as Newton-second \((N·s)\).
143. Which physical quantity represents the rate of change of linear momentum?
ⓐ. Work
ⓑ. Power
ⓒ. Force
ⓓ. Energy
Correct Answer: Force
Explanation: Newton’s second law states that the net force on a particle is equal to the time rate of change of its momentum, i.e., \(\vec{F} = \frac{d\vec{p}}{dt}\).
144. If the linear momentum of a particle doubles while its mass remains constant, its velocity will:
ⓐ. Double
ⓑ. Halve
ⓒ. Remain constant
ⓓ. Increase four times
Correct Answer: Double
Explanation: Since \(\vec{p} = m \vec{v}\), for constant \(m\), if momentum doubles, velocity must also double.
145. A truck of mass \(2000 \, kg\) moving with a velocity of \(10 \, m/s\) has linear momentum:
ⓐ. \(20000 \, kg·m/s\)
ⓑ. \(1000 \, kg·m/s\)
ⓒ. \(2 \, kg·m/s\)
ⓓ. \(5000 \, kg·m/s\)
Correct Answer: \(20000 \, kg·m/s\)
Explanation: Linear momentum \(\vec{p} = m v = 2000 \times 10 = 20000 \, kg·m/s\).
146. Which of the following statements about linear momentum is correct?
ⓐ. It is a scalar quantity
ⓑ. It always points opposite to velocity
ⓒ. It is a conserved quantity in absence of external forces
ⓓ. It has no relation with force
Correct Answer: It is a conserved quantity in absence of external forces
Explanation: Linear momentum is conserved when the net external force on a system is zero. This is the law of conservation of linear momentum.
147. Impulse is defined as:
ⓐ. Change in displacement
ⓑ. Change in velocity
ⓒ. Change in linear momentum
ⓓ. Change in acceleration
Correct Answer: Change in linear momentum
Explanation: Impulse is the product of force and time (\(J = F \Delta t\)), which equals the change in momentum of the particle.
148. Which of the following everyday situations illustrates conservation of linear momentum?
ⓐ. A book lying on a table
ⓑ. A rocket launching into space
ⓒ. A car moving with uniform speed
ⓓ. A ball kept on the ground
Correct Answer: A rocket launching into space
Explanation: The rocket expels gases backward, and by conservation of momentum, it gains forward momentum. The COM of the rocket–gas system moves as if no external force acted.
149. For a particle of constant mass, the relationship between force and momentum is:
ⓐ. \(\vec{F} = \vec{p} \cdot \vec{v}\)
ⓑ. \(\vec{F} = \frac{d\vec{p}}{dt}\)
ⓒ. \(\vec{F} = \vec{p}/t\)
ⓓ. \(\vec{F} = m\vec{p}\)
Correct Answer: \(\vec{F} = \frac{d\vec{p}}{dt}\)
Explanation: Newton’s second law in general form states that force is the rate of change of momentum. For constant mass, this reduces to \(\vec{F} = m \vec{a}\).
150. Which of the following quantities has the same dimensions as linear momentum?
ⓐ. Work
ⓑ. Torque
ⓒ. Impulse
ⓓ. Power
Correct Answer: Impulse
Explanation: Impulse is force multiplied by time. Its unit is \(N·s\), which is dimensionally the same as momentum \((kg·m/s)\). Thus, impulse equals the change in momentum.
151. The principle of conservation of linear momentum states that:
ⓐ. Momentum of a body always remains constant
ⓑ. Momentum of a system remains constant if no external force acts
ⓒ. Momentum is conserved only in elastic collisions
ⓓ. Momentum is always destroyed in inelastic collisions
Correct Answer: Momentum of a system remains constant if no external force acts
Explanation: The principle says that in an isolated system, the vector sum of momenta before and after an interaction (collision, explosion, etc.) remains the same, since internal forces cancel in pairs.
152. Two ice skaters push each other apart on a frictionless surface. What happens to their total momentum?
ⓐ. It becomes zero
ⓑ. It increases
ⓒ. It decreases
ⓓ. It remains constant
Correct Answer: It remains constant
Explanation: The forces between the skaters are internal to the system. Since there is no external force, the momentum of the system remains constant, though each skater gains momentum in opposite directions.
153. In a gun–bullet system, when the bullet is fired forward, the gun recoils backward. This demonstrates:
ⓐ. Newton’s first law
ⓑ. Newton’s third law only
ⓒ. Conservation of energy
ⓓ. Conservation of linear momentum
Correct Answer: Conservation of linear momentum
Explanation: The forward momentum of the bullet is equal and opposite to the recoil momentum of the gun, so total momentum of the gun–bullet system remains conserved.
154. If two particles of masses \(m_1\) and \(m_2\) collide in the absence of external forces, the total linear momentum before collision is:
ⓐ. Greater than after collision
ⓑ. Less than after collision
ⓒ. Equal to after collision
ⓓ. Always zero
Correct Answer: Equal to after collision
Explanation: In the absence of external forces, the total momentum before collision equals the total momentum after collision, regardless of whether the collision is elastic or inelastic.
155. A bomb explodes into three fragments in space where no external force acts. The vector sum of momenta of all fragments is:
ⓐ. Zero
ⓑ. Infinite
ⓒ. Equal to the heaviest fragment’s momentum
ⓓ. Cannot be determined
Correct Answer: Zero
Explanation: Before explosion, total momentum was zero (if the bomb was at rest). After explosion, the vector sum of all fragments’ momenta remains zero due to conservation of momentum.
156. When a moving trolley collides with a stationary trolley of equal mass and sticks to it, the velocity of the system becomes:
ⓐ. Equal to initial velocity of the moving trolley
ⓑ. Half of initial velocity of the moving trolley
ⓒ. Double the initial velocity
ⓓ. Zero
Correct Answer: Half of initial velocity of the moving trolley
Explanation: By conservation of momentum: \(m v = (m+m)V\). Solving gives \(V = v/2\). Thus, the system moves with half the initial velocity.
157. Why does a rocket accelerate upward when gases are expelled downward?
ⓐ. Because of Newton’s first law
ⓑ. Because the gases push against the ground
ⓒ. Due to conservation of linear momentum
ⓓ. Because of reduction in mass
Correct Answer: Due to conservation of linear momentum
Explanation: The momentum carried by the exhaust gases downward is balanced by the forward momentum of the rocket upward. This is conservation of momentum in action.
158. If a person jumps out of a boat on water, the boat moves backward. Which principle is involved?
ⓐ. Conservation of energy
ⓑ. Conservation of mass
ⓒ. Conservation of linear momentum
ⓓ. Newton’s law of gravitation
Correct Answer: Conservation of linear momentum
Explanation: The forward momentum of the person is balanced by the backward momentum of the boat. Thus, the system’s total momentum remains constant.
159. In elastic collisions, conservation of linear momentum holds true because:
ⓐ. Only kinetic energy is conserved
ⓑ. Internal forces cancel each other out
ⓒ. Masses become equal
ⓓ. External forces increase
Correct Answer: Internal forces cancel each other out
Explanation: In elastic collisions, internal forces (equal and opposite) cannot change the net momentum of the system. Hence, the system’s linear momentum is conserved.
160. Which of the following is an everyday example of conservation of linear momentum?
ⓐ. A satellite orbiting Earth
ⓑ. A person walking on the ground
ⓒ. A rocket taking off from Earth
ⓓ. A ball falling freely
Correct Answer: A rocket taking off from Earth
Explanation: The rocket moves upward by expelling gases downward, and the total momentum of the rocket–gas system remains conserved. This is a clear real-life demonstration of momentum conservation.
161. In an elastic collision between two bodies, which of the following quantities are conserved?
ⓐ. Only kinetic energy
ⓑ. Only linear momentum
ⓒ. Both linear momentum and kinetic energy
ⓓ. Neither linear momentum nor kinetic energy
Correct Answer: Both linear momentum and kinetic energy
Explanation: By definition, an elastic collision is one in which both linear momentum and kinetic energy are conserved. Internal forces during collision are equal and opposite, ensuring conservation of momentum, while no energy is lost as heat or deformation, ensuring conservation of kinetic energy.
162. In an inelastic collision between two bodies, which of the following is always conserved?
ⓐ. Kinetic energy only
ⓑ. Linear momentum only
ⓒ. Both kinetic energy and linear momentum
ⓓ. Neither kinetic energy nor linear momentum
Correct Answer: Linear momentum only
Explanation: In an inelastic collision, part of the kinetic energy is converted into heat, sound, or deformation, so kinetic energy is not conserved. However, linear momentum remains conserved because internal forces cancel out.
163. A ball of mass \(m\) moving with velocity \(u\) strikes an identical stationary ball elastically. What is the velocity of the first ball after collision?
ⓐ. \(u\)
ⓑ. \(0\)
ⓒ. \(\frac{u}{2}\)
ⓓ. \(-u\)
Correct Answer: \(0\)
Explanation: In an elastic head-on collision between identical masses, the moving ball transfers its entire momentum and kinetic energy to the stationary ball, coming to rest after collision.
164. A body of mass \(m_1\) moving with velocity \(u_1\) collides elastically with another body of mass \(m_2\) at rest. Which conservation laws are applied to determine final velocities?
ⓐ. Only conservation of kinetic energy
ⓑ. Only conservation of linear momentum
ⓒ. Both conservation of linear momentum and kinetic energy
ⓓ. Neither momentum nor energy conservation
Correct Answer: Both conservation of linear momentum and kinetic energy
Explanation: For elastic collisions, we use two simultaneous equations—one from momentum conservation and the other from kinetic energy conservation—to calculate the final velocities of the colliding bodies.
165. Which of the following best describes a perfectly inelastic collision?
ⓐ. Two bodies collide and rebound with no energy loss
ⓑ. Two bodies collide and stick together after collision
ⓒ. Kinetic energy is always conserved
ⓓ. Momentum is not conserved
Correct Answer: Two bodies collide and stick together after collision
Explanation: In a perfectly inelastic collision, the bodies move with a common velocity after collision. Momentum is conserved but there is maximum loss of kinetic energy.
166. Two objects collide inelastically on a frictionless surface. What happens to their total kinetic energy?
ⓐ. It increases
ⓑ. It decreases
ⓒ. It remains constant
ⓓ. It becomes zero
Correct Answer: It decreases
Explanation: In inelastic collisions, part of the initial kinetic energy is transformed into internal energy, heat, or deformation. Hence, total kinetic energy decreases, though momentum is still conserved.
167. In a head-on elastic collision between two equal masses, with one initially at rest, what happens?
ⓐ. Both masses come to rest
ⓑ. Both masses move together
ⓒ. The moving mass comes to rest, and the stationary one moves with the initial velocity
ⓓ. They exchange half their velocities
Correct Answer: The moving mass comes to rest, and the stationary one moves with the initial velocity
Explanation: For identical masses in a one-dimensional elastic collision, the velocities are exchanged. Thus, the moving one stops, and the initially stationary one moves forward.
168. Why is momentum conserved in all types of collisions, elastic or inelastic?
ⓐ. Because masses remain constant
ⓑ. Because energy is always conserved
ⓒ. Because internal forces occur in equal and opposite pairs
ⓓ. Because collisions occur in vacuum only
Correct Answer: Because internal forces occur in equal and opposite pairs
Explanation: Newton’s third law ensures that internal collision forces cancel each other. With no external force, the total momentum of the system remains constant in both elastic and inelastic collisions.
169. In which type of collision is kinetic energy not conserved but momentum is conserved?
ⓐ. Elastic collision
ⓑ. Perfectly inelastic collision
ⓒ. Super elastic collision
ⓓ. Explosion
Correct Answer: Perfectly inelastic collision
Explanation: In a perfectly inelastic collision, the bodies stick together and kinetic energy is partly converted into heat, sound, or deformation. Yet, momentum is always conserved.
170. Two cars of masses \(1000 \, \text{kg}\) and \(1200 \, \text{kg}\) collide head-on and stick together. This is an example of:
ⓐ. Elastic collision
ⓑ. Perfectly inelastic collision
ⓒ. Explosion
ⓓ. Elastic scattering
Correct Answer: Perfectly inelastic collision
Explanation: Since the cars stick together after collision, they move with a common velocity, which is the hallmark of a perfectly inelastic collision. Momentum is conserved, but kinetic energy is not.
171. According to Newton’s second law, force is related to momentum as:
ⓐ. \(\vec{F} = m \vec{v}\)
ⓑ. \(\vec{F} = \frac{d\vec{p}}{dt}\)
ⓒ. \(\vec{F} = \vec{p} \cdot \vec{v}\)
ⓓ. \(\vec{F} = m \vec{a}^2\)
Correct Answer: \(\vec{F} = \frac{d\vec{p}}{dt}\)
Explanation: The general form of Newton’s second law states that the net external force acting on a particle is equal to the time rate of change of its momentum. For constant mass, this reduces to \(\vec{F} = m \vec{a}\).
172. If the mass of a particle remains constant, the relation \(\vec{F} = \frac{d\vec{p}}{dt}\) simplifies to:
ⓐ. \(\vec{F} = m \vec{a}\)
ⓑ. \(\vec{F} = \vec{p}/t\)
ⓒ. \(\vec{F} = m \vec{v}\)
ⓓ. \(\vec{F} = \frac{\vec{v}}{a}\)
Correct Answer: \(\vec{F} = m \vec{a}\)
Explanation: For constant mass, \(\vec{p} = m \vec{v}\). Differentiating gives \(\frac{d\vec{p}}{dt} = m \frac{d\vec{v}}{dt} = m \vec{a}\).
173. Which of the following statements about force and momentum is correct?
ⓐ. Force is independent of momentum
ⓑ. Force equals the rate of change of momentum
ⓒ. Momentum is always greater than force
ⓓ. Force and momentum are unrelated quantities
Correct Answer: Force equals the rate of change of momentum
Explanation: Newton’s second law defines force in terms of momentum. This law is more general than \(F = ma\), as it also applies to variable-mass systems like rockets.
174. The impulse imparted to a body is equal to:
ⓐ. Rate of change of velocity
ⓑ. Change in linear momentum
ⓒ. Product of momentum and velocity
ⓓ. Product of force and acceleration
Correct Answer: Change in linear momentum
Explanation: Impulse \(J = F \Delta t\). From Newton’s second law, \(F = \frac{\Delta p}{\Delta t}\). Multiplying by \(\Delta t\), we get \(J = \Delta p\), i.e., impulse equals the change in momentum.
175. A cricket player lowers his hands while catching a ball. This action reduces the force on his hands because:
ⓐ. It reduces the mass of the ball
ⓑ. It reduces the velocity of the ball
ⓒ. It increases the time of momentum change
ⓓ. It eliminates momentum completely
Correct Answer: It increases the time of momentum change
Explanation: Since \(F = \frac{\Delta p}{\Delta t}\), increasing the time \(\Delta t\) during which momentum changes reduces the average force experienced.
176. If a constant force acts on a body of mass \(m\) for a time \(t\), the change in momentum is:
ⓐ. \(F \cdot t\)
ⓑ. \(F/m\)
ⓒ. \(m/F\)
ⓓ. \(F \cdot v\)
Correct Answer: \(F \cdot t\)
Explanation: From impulse–momentum theorem, the change in momentum \(\Delta p = F \cdot t\), provided force is constant during the interaction.
177. The braking force applied to a moving car brings it to rest. What happens to its momentum?
ⓐ. It increases
ⓑ. It decreases to zero
ⓒ. It becomes infinite
ⓓ. It remains constant
Correct Answer: It decreases to zero
Explanation: The external braking force changes the momentum of the car. Since final velocity is zero, momentum reduces from its initial value to zero.
178. A rocket moving in space expels gases backward. This is explained by:
ⓐ. \(\vec{F} = m \vec{a}\) only
ⓑ. Conservation of angular momentum
ⓒ. Force as the rate of change of momentum
ⓓ. Law of gravitation
Correct Answer: Force as the rate of change of momentum
Explanation: As gases are expelled backward with momentum, the rocket gains equal and opposite momentum forward. The thrust force arises from the rate of change of momentum of the exhaust gases.
179. Which of the following graphs correctly represents the relationship between force and momentum?
ⓐ. Force is always proportional to momentum itself
ⓑ. Force is proportional to the time derivative of momentum
ⓒ. Force is inversely proportional to momentum
ⓓ. Force is independent of momentum
Correct Answer: Force is proportional to the time derivative of momentum
Explanation: By Newton’s second law, force depends on how fast momentum changes with time, not on momentum itself.
180. Why is Newton’s second law expressed as \(\vec{F} = \frac{d\vec{p}}{dt}\) more general than \(\vec{F} = m \vec{a}\)?
ⓐ. Because it applies only for uniform motion
ⓑ. Because it applies even when mass varies with time
ⓒ. Because it excludes rotational systems
ⓓ. Because it avoids use of velocity
Correct Answer: Because it applies even when mass varies with time
Explanation: In cases like rocket propulsion, mass changes due to fuel consumption. Here, \(F = \frac{d\vec{p}}{dt}\) correctly describes dynamics, while \(F = ma\) applies only when mass is constant.
181. The vector product of two vectors \(\vec{A}\) and \(\vec{B}\) is defined as:
ⓐ. \(\vec{A} \cdot \vec{B}\)
ⓑ. \(|\vec{A}| |\vec{B}| \cos\theta\)
ⓒ. \(\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin\theta \, \hat{n}\)
ⓓ. \(\vec{A} + \vec{B}\)
Correct Answer: \(\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin\theta \, \hat{n}\)
Explanation: The vector product (cross product) of two vectors is a vector whose magnitude is \(|\vec{A}| |\vec{B}| \sin\theta\) and direction is perpendicular to the plane of \(\vec{A}\) and \(\vec{B}\), given by the right-hand rule.
182. The direction of the vector product \(\vec{A} \times \vec{B}\) is determined by:
ⓐ. Left-hand rule
ⓑ. Right-hand rule
ⓒ. Fleming’s rule
ⓓ. Newton’s rule
Correct Answer: Right-hand rule
Explanation: The right-hand screw rule or right-hand thumb rule gives the direction of \(\vec{A} \times \vec{B}\). Curl the fingers of your right hand from \(\vec{A}\) to \(\vec{B}\); the thumb points in the direction of the cross product.
183. Which of the following is true about the vector product of two vectors?
ⓐ. It is always a scalar
ⓑ. It is always parallel to the vectors
ⓒ. It is always perpendicular to the plane of the vectors
ⓓ. It is always zero
Correct Answer: It is always perpendicular to the plane of the vectors
Explanation: By definition, the cross product yields a vector perpendicular to both \(\vec{A}\) and \(\vec{B}\). Its direction is unique up to sign, determined by the right-hand rule.
184. If two vectors are parallel, their vector product is:
ⓐ. Maximum
ⓑ. Zero
ⓒ. Negative
ⓓ. Perpendicular to both vectors
Correct Answer: Zero
Explanation: Since \(\sin\theta = 0\) when vectors are parallel (\(\theta = 0^\circ\) or \(180^\circ\)), the cross product vanishes. Thus, \(\vec{A} \times \vec{B} = 0\).
185. If two vectors are perpendicular, the magnitude of their cross product is:
ⓐ. Zero
ⓑ. Equal to the product of their magnitudes
ⓒ. Half the product of their magnitudes
ⓓ. Double the product of their magnitudes
Correct Answer: Equal to the product of their magnitudes
Explanation: For perpendicular vectors, \(\theta = 90^\circ\), so \(\sin\theta = 1\). Thus, \(|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}|\).
186. Which of the following properties is correct for vector product?
ⓐ. \(\vec{A} \times \vec{B} = \vec{B} \times \vec{A}\)
ⓑ. \(\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})\)
ⓒ. \(\vec{A} \times \vec{B} = \vec{0}\) always
ⓓ. \(\vec{A} \times \vec{B} = \vec{A} \cdot \vec{B}\)
Correct Answer: \(\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})\)
Explanation: The cross product is anti-commutative. Reversing the order of multiplication reverses the direction of the result.
187. The magnitude of the vector product represents:
ⓐ. The area of a triangle formed by the vectors
ⓑ. The volume of a cube formed by the vectors
ⓒ. The area of a parallelogram formed by the vectors
ⓓ. The scalar projection of one vector on the other
Correct Answer: The area of a parallelogram formed by the vectors
Explanation: Since \(|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta\), it equals the area of a parallelogram with sides \(\vec{A}\) and \(\vec{B}\).
188. Which distributive property is satisfied by the vector product?
ⓐ. \(\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}\)
ⓑ. \(\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}\)
ⓒ. \(\vec{A} \times (\vec{B} + \vec{C}) = 0\)
ⓓ. \(\vec{A} \times (\vec{B} + \vec{C}) = (\vec{B} + \vec{C}) \times \vec{A}\)
Correct Answer: \(\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}\)
Explanation: The cross product is distributive over vector addition but remains anti-commutative, meaning order reversal changes the sign.
189. Which of the following statements is NOT true about vector product?
ⓐ. It is non-commutative
ⓑ. It is distributive over addition
ⓒ. It always produces a scalar
ⓓ. Its direction is given by right-hand rule
Correct Answer: It always produces a scalar
Explanation: The cross product is not a scalar but a vector quantity. It gives both magnitude and direction, unlike the scalar (dot) product.
190. The cross product of a vector with itself is always:
ⓐ. Zero
ⓑ. Equal to its magnitude squared
ⓒ. Undefined
ⓓ. Equal to one
Correct Answer: Zero
Explanation: Since the angle between a vector and itself is \(0^\circ\), \(\sin 0 = 0\). Hence, \(\vec{A} \times \vec{A} = 0\).
191. The geometric interpretation of the magnitude of \(\vec{A} \times \vec{B}\) is:
ⓐ. The area of a triangle formed by \(\vec{A}\) and \(\vec{B}\)
ⓑ. The length of the diagonal of the parallelogram
ⓒ. The area of the parallelogram formed by \(\vec{A}\) and \(\vec{B}\)
ⓓ. The dot product of the vectors
Correct Answer: The area of the parallelogram formed by \(\vec{A}\) and \(\vec{B}\)
Explanation: Geometrically, \(|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta\) represents the area of the parallelogram constructed on the two vectors as adjacent sides.
192. If the area of a triangle with sides represented by vectors \(\vec{A}\) and \(\vec{B}\) is required, it is given by:
ⓐ. \(|\vec{A} \cdot \vec{B}|/2\)
ⓑ. \(|\vec{A} \times \vec{B}|/2\)
ⓒ. \(|\vec{A} + \vec{B}|/2\)
ⓓ. \(|\vec{A}| |\vec{B}|/2\)
Correct Answer: \(|\vec{A} \times \vec{B}|/2\)
Explanation: The magnitude of the cross product gives the area of the parallelogram formed by the vectors. Since a triangle is half of that parallelogram, its area is \(|\vec{A} \times \vec{B}|/2\).
193. The direction of \(\vec{A} \times \vec{B}\) geometrically represents:
ⓐ. Along the diagonal of the parallelogram formed by \(\vec{A}\) and \(\vec{B}\)
ⓑ. Perpendicular to the plane of \(\vec{A}\) and \(\vec{B}\)
ⓒ. Opposite to the vector \(\vec{A}\)
ⓓ. Along the line of action of \(\vec{B}\)
Correct Answer: Perpendicular to the plane of \(\vec{A}\) and \(\vec{B}\)
Explanation: The cross product produces a vector perpendicular to both \(\vec{A}\) and \(\vec{B}\). Its exact orientation is given by the right-hand rule.
194. When vectors \(\vec{A}\) and \(\vec{B}\) are parallel, the geometric interpretation of \(\vec{A} \times \vec{B}\) is:
ⓐ. Maximum possible area of parallelogram
ⓑ. Zero area of parallelogram
ⓒ. Minimum possible area but not zero
ⓓ. Undefined
Correct Answer: Zero area of parallelogram
Explanation: If vectors are parallel, \(\theta = 0^\circ\) or \(180^\circ\), so \(\sin \theta = 0\). The parallelogram collapses into a line, and area becomes zero.
195. When vectors \(\vec{A}\) and \(\vec{B}\) are perpendicular, the geometric interpretation of their cross product is:
ⓐ. Area of a parallelogram = \(|\vec{A}| |\vec{B}|\)
ⓑ. Zero area
ⓒ. A scalar equal to 1
ⓓ. A vector with magnitude less than dot product
Correct Answer: Area of a parallelogram = \(|\vec{A}| |\vec{B}|\)
Explanation: For \(\theta = 90^\circ\), \(\sin\theta = 1\). Hence, \(|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}|\), the maximum possible area.
196. The geometric interpretation of the unit vector \(\hat{n}\) in \(\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin\theta \, \hat{n}\) is:
ⓐ. A vector parallel to \(\vec{A}\)
ⓑ. A vector parallel to \(\vec{B}\)
ⓒ. A vector normal to the plane of \(\vec{A}\) and \(\vec{B}\)
ⓓ. A vector along the bisector of \(\vec{A}\) and \(\vec{B}\)
Correct Answer: A vector normal to the plane of \(\vec{A}\) and \(\vec{B}\)
Explanation: The unit vector \(\hat{n}\) specifies the direction of the cross product, perpendicular to both vectors, consistent with the right-hand rule.
197. If \(|\vec{A} \times \vec{B}| = 0\), geometrically this means:
ⓐ. The parallelogram has maximum area
ⓑ. The two vectors are perpendicular
ⓒ. The two vectors are collinear
ⓓ. The vectors have equal magnitudes
Correct Answer: The two vectors are collinear
Explanation: Zero cross product magnitude occurs when vectors are parallel or anti-parallel, i.e., collinear, so the area of parallelogram is zero.
198. The geometric meaning of the magnitude of \(\vec{A} \times \vec{B}\) is equivalent to:
ⓐ. Projection of \(\vec{A}\) on \(\vec{B}\)
ⓑ. Projection of \(\vec{B}\) on \(\vec{A}\)
ⓒ. Base times height of the parallelogram formed by \(\vec{A}\) and \(\vec{B}\)
ⓓ. Sum of squares of magnitudes of vectors
Correct Answer: Base times height of the parallelogram formed by \(\vec{A}\) and \(\vec{B}\)
Explanation: Geometrically, \(|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta\), which equals the base (\(|\vec{A}|\)) times the height (\(|\vec{B}| \sin\theta\)) of the parallelogram.
199. The geometric interpretation of vector product helps in finding:
ⓐ. The length of a vector
ⓑ. The angle between two vectors only
ⓒ. The area enclosed by two vectors and the plane normal
ⓓ. The scalar magnitude of one vector
Correct Answer: The area enclosed by two vectors and the plane normal
Explanation: The vector product provides both the area of the parallelogram (magnitude) and the orientation of the plane (direction).
200. Which of the following best represents the geometric interpretation of the vector product in 3D space?
ⓐ. A line segment parallel to one vector
ⓑ. A scalar proportional to the cosine of the angle
ⓒ. A directed area vector perpendicular to the plane of the two vectors
ⓓ. A dimensionless constant
Correct Answer: A directed area vector perpendicular to the plane of the two vectors
Explanation: The cross product geometrically represents an area vector—magnitude equals the parallelogram’s area, direction perpendicular to the plane, giving complete geometric meaning in 3D.
The chapter System of Particles and Rotational Motion in Class 11 Physics is a key topic of the NCERT/CBSE syllabus and is often tested in board exams, JEE, NEET, and state-level entrance exams. It covers vital principles such as motion of center of mass, equilibrium of rigid bodies, torque, and the laws of rotation. The complete chapter contains 570 MCQs with step-by-step explanations, divided into 6 systematic parts. This section provides the second set of 100 MCQs with solutions, perfect for practicing more advanced problems and enhancing problem-solving skills.
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