1. Which description best matches a system of particles in mechanics?
ⓐ. A rigid body restricted to rotational motion only
ⓑ. A single particle chosen for point-mass analysis
ⓒ. A point fixed permanently at the origin
ⓓ. A collection of particles studied together
Correct Answer: A collection of particles studied together
Explanation: A system of particles means a group of particles considered as one chosen system. The particles may interact with one another, and forces from outside the system may also act on them. This idea lets us describe the motion of many-particle bodies without following every particle separately in every situation. For translational motion, the centre of mass often represents the system in a compact way. A single-particle model is only a special simpler case, while a system can contain many particles with different positions and velocities.
2. A rigid body is best idealised as a body in which
ⓐ. all particles move only along straight paths
ⓑ. distances between particle pairs remain fixed
ⓒ. the body has zero mass and zero size
ⓓ. every particle remains fixed at one position
Correct Answer: distances between particle pairs remain fixed
Explanation: A rigid body is an ideal body whose shape and size do not change during motion. This means the separation between any two particles of the body remains fixed. The body as a whole may translate, rotate, or do both, but its internal geometry is treated as unchanged. Real bodies can deform slightly, yet the rigid-body approximation works well when the deformation is too small to affect the motion being studied. Rigidity is about fixed relative distances, not about the body being motionless.
3. A block sliding on a smooth horizontal table without turning is an example mainly of
ⓐ. circular motion of every particle about the same axis
ⓑ. pure rotational motion about its centre
ⓒ. deformation without motion
ⓓ. pure translational motion
Correct Answer: pure translational motion
Explanation: In pure translational motion, all particles of the body have the same displacement during the same time interval. A sliding block that does not turn keeps its orientation unchanged while every point shifts in the same direction. No point of the block is revolving around an axis of the block. Rotation would require the body to change its orientation or for its particles to move in circles around an axis. The key feature here is common displacement of all points, not the shape of the path alone.
4. Which idea is closest to the basic meaning of centre of mass?
ⓐ. A mass-weighted representative point for translational motion
ⓑ. The point that must always lie inside the material of the body
ⓒ. The point where all external forces must always act
ⓓ. The geometrical centre of every body
Correct Answer: A mass-weighted representative point for translational motion
Explanation: The centre of mass is a representative point whose position depends on how mass is distributed in a system. It is called mass-weighted because heavier parts influence its position more strongly than lighter parts. For many purposes, the translational motion of a system can be described as the motion of its centre of mass. It is not necessarily the same as the geometrical centre unless the body has suitable uniformity and symmetry. The centre of mass may even lie at a point where no material is present, such as the centre of a uniform ring.
5. In rotational motion about a fixed axis, the particles of a rigid body generally
ⓐ. remain at the same point in space
ⓑ. move with no relation to one another
ⓒ. move in circles whose centres lie on the axis
ⓓ. lose their fixed separation from neighbouring particles
Correct Answer: move in circles whose centres lie on the axis
Explanation: When a rigid body rotates about a fixed axis, each particle not lying on the axis follows a circular path. The centre of that circular path lies on the axis of rotation. Particles at different distances from the axis trace circles of different radii. The body remains rigid because the relative distances between its particles do not change during this motion. Rotation is identified by change of orientation about an axis, not by the body simply shifting from one place to another.
6. Match the basic quantity or symbol with its usual meaning.
| Column I | Column II |
|---|
| P. \(\vec{r}\) | 1. Position vector of a particle |
| Q. \(M\) | 2. Total mass of the system |
| R. \(\vec{v}_{\text{CM}}\) | 3. Velocity of the centre of mass |
| S. \(\vec{\omega}\) | 4. Angular velocity |
ⓐ. P-2, Q-1, R-3, S-4
ⓑ. P-1, Q-2, R-4, S-3
ⓒ. P-1, Q-2, R-3, S-4
ⓓ. P-3, Q-2, R-1, S-4
Correct Answer: P-1, Q-2, R-3, S-4
Explanation: The symbol \(\vec{r}\) commonly represents the position vector of a particle measured from a chosen origin. The symbol \(M\) is used for the total mass of a system, especially when many particles are being treated together. The notation \(\vec{v}_{\text{CM}}\) represents the velocity of the centre of mass. The vector \(\vec{\omega}\) represents angular velocity, whose direction is associated with the axis of rotation. The vector arrow matters for quantities such as \(\vec{r}\), \(\vec{v}_{\text{CM}}\), and \(\vec{\omega}\), while \(M\) is a scalar mass.
7. The coordinate of the centre of mass is a position coordinate, so its SI unit is
ⓐ. \(\text{rad s}^{-1}\)
ⓑ. \(\text{kg}\)
ⓒ. \(\text{s}\)
ⓓ. \(\text{m}\)
Correct Answer: \(\text{m}\)
Explanation: A centre-of-mass coordinate tells where the centre of mass is located along an axis. Since it is a position coordinate, it has the same unit as length. In SI, length is measured in \(\text{m}\). The unit \(\text{kg}\) belongs to mass, \(\text{s}\) belongs to time, and \(\text{rad s}^{-1}\) belongs to angular velocity. The centre of mass may depend on masses, but its coordinate still measures position.
8. Study the row entries and identify the fully suitable pairing.
| Row | Quantity | Usual unit |
|---|
| P | Total mass \(M\) | \(\text{kg}\) |
| Q | Time \(t\) | \(\text{m}\) |
| R | Angular velocity \(\omega\) | \(\text{N m}\) |
| S | Position coordinate \(x\) | \(\text{rad s}^{-1}\) |
ⓐ. Row Q
ⓑ. Row S
ⓒ. Row R
ⓓ. Row P
Correct Answer: Row P
Explanation: The total mass \(M\) of a system is measured in \(\text{kg}\), so row P gives a suitable pairing. Time \(t\) is measured in \(\text{s}\), not in \(\text{m}\). Angular velocity \(\omega\) is measured in \(\text{rad s}^{-1}\), while \(\text{N m}\) is the unit of torque. A position coordinate such as \(x\) is measured in \(\text{m}\), not in \(\text{rad s}^{-1}\). Unit matching is useful because it separates position, mass, time, rotation rate, and turning effect.
9. Consider the following statements about the particle model and the rigid-body model.
I. A particle model ignores the size and shape of the body.
II. A rigid-body model allows rotation while keeping relative distances fixed.
III. A rigid body must always be treated as a single point.
ⓐ. I and II only
ⓑ. II and III only
ⓒ. I and III only
ⓓ. I, II, and III
Correct Answer: I and II only
Explanation: A particle model treats a body as if its size and shape are not important for the question being studied. A rigid-body model keeps the size and shape important, but assumes the body does not deform. Such a body can translate, rotate, or perform a combination of both motions. Statement III is not suitable because a rigid body is extended and need not be reduced to a single point for rotational study. The particle model is useful for pure translation, while the rigid-body model is needed when orientation and rotation matter.
10. A rotating ceiling fan is considered a rigid body in elementary rotational motion because
ⓐ. its mass becomes zero during rotation
ⓑ. the blades keep fixed shape while rotating
ⓒ. all blade points have the same linear speed
ⓓ. its axis of rotation is absent
Correct Answer: the blades keep fixed shape while rotating
Explanation: A ceiling fan is treated as a rigid body when the distances between particles of its blades are assumed to remain unchanged. The blades rotate about a fixed axis, but their shape is not considered to deform appreciably. Points at different distances from the axis generally have different linear speeds, so equal linear speed does not describe rigid-body behaviour. The fan certainly has an axis of rotation passing through its centre. The rigid-body approximation lets us study rotation without tracking small elastic bending of the blades.
11. A wheel rolls forward on a horizontal road without slipping. At the simplest classification level, its motion is a combination of
ⓐ. rotation only, with no forward motion
ⓑ. translation only, with no rotation
ⓒ. translation of its centre with rotation about it
ⓓ. deformation of the wheel with no regular rolling
Correct Answer: translation of its centre with rotation about it
Explanation: A rolling wheel moves forward as a whole, so its centre has translational motion. At the same time, the wheel turns about its central axis, so it also has rotational motion. This combination is different from a block sliding without turning, which is mainly translational. It is also different from a wheel spinning in place, where the centre does not move forward. Rolling motion becomes clearer only when both translation of the centre and rotation about the centre are kept together.
12. The total mass symbol \(M\) for a system of particles represents
ⓐ. the mass of only the fastest particle
ⓑ. the sum of all particle masses in the system
ⓒ. the algebraic sum of position coordinates
ⓓ. the distance of the centre of mass from the origin
Correct Answer: the sum of all particle masses in the system
Explanation: For a system of particles, \(M\) denotes the total mass included in the chosen system. If the particles have masses \(m_1\), \(m_2\), \(m_3\), and so on, then \(M=m_1+m_2+m_3+\cdots\). This is not a position coordinate and does not depend on which particle is moving fastest. The system boundary decides which particles are included in \(M\). This notation is important because centre-of-mass formulas divide the mass-weighted position sum by the total mass.
13. For three particles of masses \(2\,\text{kg}\), \(3\,\text{kg}\), and \(5\,\text{kg}\) included in one system, what is the total mass \(M\)?
ⓐ. \(10\,\text{kg}\)
ⓑ. \(5\,\text{kg}\)
ⓒ. \(8\,\text{kg}\)
ⓓ. \(30\,\text{kg}\)
Correct Answer: \(10\,\text{kg}\)
Explanation: \( \textbf{Given masses:} \) \(m_1=2\,\text{kg}\), \(m_2=3\,\text{kg}\), and \(m_3=5\,\text{kg}\).
\( \textbf{Required quantity:} \) Total mass \(M\) of the chosen system.
\( \textbf{Relevant relation:} \)
\[
M=m_1+m_2+m_3
\]
\( \textbf{Why this relation applies:} \) Total mass is found by adding the masses of all particles included in the system.
\( \textbf{Substitution:} \)
\[
M=2\,\text{kg}+3\,\text{kg}+5\,\text{kg}
\]
\( \textbf{Addition:} \)
\[
M=10\,\text{kg}
\]
\( \textbf{Unit check:} \) Each term is a mass in \(\text{kg}\), so the final unit remains \(\text{kg}\).
\( \textbf{Final answer:} \) The total mass of the system is \(10\,\text{kg}\).
14. Use the arrangement described below. Two unequal masses are placed on a light horizontal line, with the heavier mass on the right and the lighter mass on the left. Without calculating, the centre of mass of the two-mass system will lie
ⓐ. exactly midway between the two masses
ⓑ. closer to the heavier mass
ⓒ. exactly at the left mass
ⓓ. outside the line segment on the lighter side
Correct Answer: closer to the heavier mass
Explanation: The centre of mass is a mass-weighted position, so the heavier mass has a stronger influence on its location. For two masses placed on a line, the centre of mass lies somewhere between them if both masses are positive and separated. It is exactly at the midpoint only when the two masses are equal. Since the right mass is heavier here, the representative point shifts toward the right side. This gives the physical meaning before using the formula \(X_{\text{CM}}=\frac{m_1x_1+m_2x_2}{m_1+m_2}\).
15. The expression \(\vec{R}_{\text{CM}}\) most naturally denotes
ⓐ. the radius vector of a circular path only
ⓑ. the rotational kinetic energy of a body
ⓒ. the position vector of the centre of mass
ⓓ. the torque acting on the chosen system
Correct Answer: the position vector of the centre of mass
Explanation: The symbol \(\vec{R}_{\text{CM}}\) is commonly used for the position vector of the centre of mass of a system. The subscript \(\text{CM}\) indicates that the quantity belongs to the centre of mass. The vector arrow shows that position has both magnitude and direction relative to a chosen origin. Rotational kinetic energy is usually written as \(K_{\text{rot}}\), while torque is represented by \(\vec{\tau}\). The notation \(\vec{R}_{\text{CM}}\) is a system-level position, not simply the radius of one particle’s circular path.
16. A body moves so that every line drawn inside it remains parallel to its initial direction throughout the motion. This observation mainly indicates
ⓐ. rotation about a fixed axis
ⓑ. change in shape due to deformation
ⓒ. zero velocity of the body
ⓓ. translational motion
Correct Answer: translational motion
Explanation: If every line in a body remains parallel to its original direction, the body is not changing its orientation. That is the signature of translational motion. The body may move along a straight or curved path, but all its particles have equal displacement during the same time interval in pure translation. Rotation would make internal lines turn through an angle, changing their direction in space. Translation is about common displacement of all points, not about the body necessarily moving with constant velocity.
17. In the notation used for rotational motion, \(I\) is introduced as the symbol for
ⓐ. linear momentum
ⓑ. centre-of-mass velocity
ⓒ. moment of inertia
ⓓ. angular displacement
Correct Answer: moment of inertia
Explanation: The symbol \(I\) denotes moment of inertia in rotational motion. It is the rotational analogue of mass in the sense that it measures resistance to angular acceleration about a specified axis. Unlike mass, \(I\) depends on how mass is distributed relative to the chosen axis. Its SI unit is \(\text{kg m}^2\), which already shows that both mass and distance from the axis are involved. This symbol is central in rotational dynamics and rotational kinetic energy.
18. A rotating door, a spinning wheel, and a balanced beam are useful early examples because all of them involve
ⓐ. point-particle motion where body size has no role
ⓑ. systems on which forces cannot act
ⓒ. extended bodies where turning effects matter
ⓓ. bodies whose total mass must be zero
Correct Answer: extended bodies where turning effects matter
Explanation: These examples cannot be fully described by treating the body as a single point in every situation. A rotating door has an axis near its hinge, a spinning wheel rotates about its axle, and a balanced beam depends on how forces act at different positions. Their size, shape, and orientation are relevant to the motion or balance. Such examples motivate the need for rigid-body ideas after single-particle mechanics. They also show why quantities such as torque, angular velocity, and moment of inertia are needed for extended bodies.
19. A system boundary is chosen around two carts connected by a compressed spring. If both carts are included in the system, the spring forces between the carts are classified as
ⓐ. internal forces between parts of the system
ⓑ. rotational forces because the carts separate
ⓒ. external forces because they change speeds
ⓓ. zero forces because springs cannot act inside
Correct Answer: internal forces between parts of the system
Explanation: A force is called internal or external only after the system has been chosen. When both carts are included in the same system, the spring force exerted by one cart on the other is a force between parts of that system. Such forces may change the motion of individual particles or bodies inside the system, but they are still internal forces for the system as a whole. If one cart alone were chosen as the system, the spring force on that cart would be external. The classification depends on the chosen boundary, not on whether the force is large or small.
20. A stone tied to a string is whirled in a circle. If the chosen system is only the stone, the tension exerted by the string on the stone is
ⓐ. an internal force
ⓑ. an external force
ⓒ. absent because the motion is circular
ⓓ. a force that must be ignored in system analysis
Correct Answer: an external force
Explanation: The chosen system here contains only the stone. The string is outside that chosen system, so the force exerted by the string on the stone comes from outside the system. Therefore, the tension is an external force for the stone-only system. If the system were redefined to include both stone and string, the classification of some interactions would change. Internal and external are system-dependent labels, so the boundary must be fixed before classifying forces.
