221. Angular velocity of a particle moving in a circle of radius \(r\) with linear speed \(v\) is defined as:
ⓐ. \(\omega = vr\)
ⓑ. \(\omega = \frac{r}{v}\)
ⓒ. \(\omega = \frac{v}{r}\)
ⓓ. \(\omega = \frac{1}{vr}\)
Correct Answer: \(\omega = \frac{v}{r}\)
Explanation: Angular velocity \(\omega\) is the rate of change of angular displacement with time. For circular motion, \(v = \omega r\), hence \(\omega = \frac{v}{r}\).
222. What is the SI unit of angular velocity?
ⓐ. radian per second (\(\text{rad s}^{-1}\))
ⓑ. meter per second (\(\text{m s}^{-1}\))
ⓒ. degree per second (\(^\circ \text{s}^{-1}\))
ⓓ. hertz (Hz)
Correct Answer: radian per second (\(\text{rad s}^{-1}\))
Explanation: Angular velocity is measured in radians per unit time. The SI unit is radian per second, which indicates how many radians the object sweeps in one second.
223. If a wheel makes 120 revolutions per minute, what is its angular velocity in radians per second?
ⓐ. \(4\pi \, \text{rad/s}\)
ⓑ. \(2\pi \, \text{rad/s}\)
ⓒ. \(8\pi \, \text{rad/s}\)
ⓓ. \(6\pi \, \text{rad/s}\)
Correct Answer: \(4\pi \, \text{rad/s}\)
Explanation: 120 revolutions per minute = 2 revolutions per second. Since one revolution = \(2\pi\) radians, angular velocity = \(2 \times 2\pi = 4\pi \, \text{rad/s}\).
224. Which of the following best defines angular velocity?
ⓐ. Rate of change of displacement
ⓑ. Rate of change of angular displacement
ⓒ. Rate of change of linear momentum
ⓓ. Rate of change of torque
Correct Answer: Rate of change of angular displacement
Explanation: Angular velocity is defined as the change in angular displacement with respect to time. It describes how quickly an object rotates about a fixed axis.
225. A body rotates through an angle of \(\pi\) radians in 2 seconds. What is its average angular velocity?
ⓐ. \(\pi \, \text{rad/s}\)
ⓑ. \(\frac{\pi}{2} \, \text{rad/s}\)
ⓒ. \(2\pi \, \text{rad/s}\)
ⓓ. \(\frac{2}{\pi} \, \text{rad/s}\)
Correct Answer: \(\frac{\pi}{2} \, \text{rad/s}\)
Explanation: Average angular velocity = \(\frac{\Delta \theta}{\Delta t} = \frac{\pi}{2} \, \text{rad/s}\). It shows how much angle is covered per second during rotation.
226. If angular displacement is represented by \(\theta = 2t^2\) (in radians), what is the instantaneous angular velocity at \(t = 3 \, \text{s}\)?
ⓐ. 6 rad/s
ⓑ. 8 rad/s
ⓒ. 10 rad/s
ⓓ. 12 rad/s
Correct Answer: 12 rad/s
Explanation: Instantaneous angular velocity is \(\omega = \frac{d\theta}{dt} = \frac{d(2t^2)}{dt} = 4t\). At \(t = 3\), \(\omega = 4 \times 3 = 12 \, \text{rad/s} \).
227. If the angular velocity of a rotating fan blade is constant, which of the following statements is true?
ⓐ. Angular acceleration is non-zero
ⓑ. Angular acceleration is zero
ⓒ. Linear velocity is zero
ⓓ. Angular displacement is zero
Correct Answer: Angular acceleration is zero
Explanation: Constant angular velocity means no change in angular velocity with time. Thus, angular acceleration is zero, though the fan still has angular displacement and linear velocity at each point.
228. What is the relationship between frequency \(f\) and angular velocity \(\omega\)?
ⓐ. \(\omega = f\)
ⓑ. \(\omega = 2\pi f\)
ⓒ. \(\omega = \frac{f}{2\pi}\)
ⓓ. \(\omega = \frac{1}{f}\)
Correct Answer: \(\omega = 2\pi f\)
Explanation: One complete revolution corresponds to \(2\pi\) radians. Thus, if frequency is \(f\) revolutions per second, angular velocity is \(\omega = 2\pi f\).
229. If a point on a wheel has a linear speed of 10 m/s at a distance of 2 m from the axis, what is the angular velocity of the wheel?
ⓐ. 2 rad/s
ⓑ. 4 rad/s
ⓒ. 5 rad/s
ⓓ. 20 rad/s
Correct Answer: 5 rad/s
Explanation: Angular velocity is \(\omega = \frac{v}{r} = \frac{10}{2} = 5 \, \text{rad/s}\). This shows the connection between linear and angular velocity.
230. Which of the following is a vector quantity associated with angular velocity?
ⓐ. It has only magnitude, not direction
ⓑ. It has both magnitude and direction along the axis of rotation
ⓒ. It is scalar because it measures angular displacement per unit time
ⓓ. It is always parallel to linear velocity
Correct Answer: It has both magnitude and direction along the axis of rotation
Explanation: Angular velocity is a vector. Its magnitude shows how fast the body rotates, and its direction is along the axis of rotation, determined by the right-hand rule.
231. Which equation correctly relates angular velocity \(\omega\), linear velocity \(v\), and radius \(r\) of circular motion?
ⓐ. \(v = \frac{\omega}{r}\)
ⓑ. \(v = \omega r\)
ⓒ. \(v = \frac{r}{\omega}\)
ⓓ. \(v = \omega + r\)
Correct Answer: \(v = \omega r\)
Explanation: In circular motion, linear velocity is directly proportional to angular velocity and radius. The formula is \(v = \omega r\), which shows that for the same \(\omega\), a larger radius produces greater linear speed.
232. If the angular velocity of a wheel is \(10 \, \text{rad/s}\) and its radius is \(0.5 \, \text{m}\), what is the linear velocity of a point on the rim?
ⓐ. 2.5 m/s
ⓑ. 5 m/s
ⓒ. 10 m/s
ⓓ. 20 m/s
Correct Answer: 5 m/s
Explanation: Using \(v = \omega r\), we get \(v = 10 \times 0.5 = 5 \, \text{m/s}\). The point on the rim moves with this linear speed.
233. For a body in uniform circular motion, what is the relationship between the directions of angular velocity vector \(\omega\) and linear velocity \(v\)?
ⓐ. Both are in the same direction
ⓑ. Both are opposite
ⓒ. \(\omega\) is along the axis of rotation, while \(v\) is tangent to the circle
ⓓ. Both are radial
Correct Answer: \(\omega\) is along the axis of rotation, while \(v\) is tangent to the circle
Explanation: Linear velocity is tangential to the circular path, while angular velocity vector lies along the axis of rotation, perpendicular to the plane of motion (right-hand rule).
234. If a car moves in a circle of radius \(50 \, \text{m}\) with an angular velocity of \(0.2 \, \text{rad/s}\), what is its linear velocity?
ⓐ. 5 m/s
ⓑ. 10 m/s
ⓒ. 15 m/s
ⓓ. 20 m/s
Correct Answer: 5 m/s
Explanation: Linear velocity is given by \(v = \omega r = 0.2 \times 50 = 10 \, \text{m/s}\). Correction: Wait, recalc → \(0.2 \times 50 = 10\). So the correct answer is 10 m/s, not 5 m/s. Answer: B. 10 m/s.
235. If the radius of a rotating disc is doubled while keeping angular velocity constant, what happens to the linear velocity of a point on its rim?
ⓐ. It halves
ⓑ. It remains the same
ⓒ. It doubles
ⓓ. It becomes zero
Correct Answer: It doubles
Explanation: Since \(v = \omega r\), if \(\omega\) remains constant and \(r\) doubles, then \(v\) also doubles.
236. A stone tied to a string is whirled in a horizontal circle. If the string breaks, what happens to the stone’s motion immediately?
ⓐ. It stops at once
ⓑ. It moves radially inward
ⓒ. It moves radially outward
ⓓ. It moves tangentially to the circle
Correct Answer: It moves tangentially to the circle
Explanation: Linear velocity is always tangent to the circle in circular motion. If the centripetal force vanishes (string breaks), the stone moves tangentially in a straight line due to inertia.
237. The wheels of a train have an angular velocity of \(20 \, \text{rad/s}\) and radius \(0.6 \, \text{m}\). What is the train’s linear speed?
ⓐ. 10 m/s
ⓑ. 12 m/s
ⓒ. 15 m/s
ⓓ. 18 m/s
Correct Answer: 12 m/s
Explanation: \(v = \omega r = 20 \times 0.6 = 12 \, \text{m/s}\). This is the train’s speed along the track.
238. What is the effect on angular velocity when linear velocity is kept constant and the radius of the circular path is doubled?
ⓐ. Angular velocity doubles
ⓑ. Angular velocity halves
ⓒ. Angular velocity becomes zero
ⓓ. Angular velocity remains constant
Correct Answer: Angular velocity halves
Explanation: Since \(\omega = \frac{v}{r}\), for constant \(v\), if radius doubles, angular velocity is halved.
239. Two wheels of radii \(0.2 \, \text{m}\) and \(0.4 \, \text{m}\) are rotating at the same angular velocity. Which statement is correct?
ⓐ. Both wheels have the same linear velocity
ⓑ. The smaller wheel has greater linear velocity
ⓒ. The larger wheel has greater linear velocity
ⓓ. Both wheels have zero linear velocity
Correct Answer: The larger wheel has greater linear velocity
Explanation: Linear velocity depends on radius: \(v = \omega r\). For the same angular velocity, the larger wheel covers more distance in the same time, hence higher linear velocity.
240. A rotating ceiling fan blade has an angular velocity of \(30 \, \text{rad/s}\). If the tip of the blade is 0.4 m from the axis, what is the tip’s linear speed?
ⓐ. 6 m/s
ⓑ. 9 m/s
ⓒ. 12 m/s
ⓓ. 15 m/s
Correct Answer: 12 m/s
Explanation: Using \(v = \omega r\), we get \(v = 30 \times 0.4 = 12 \, \text{m/s}\). The blade tip moves with this tangential velocity.
