Free Rotational and Circular Motion MCQs with Answers

1. One radian is defined as the angle subtended at the centre of a circle by an arc whose length is

• A. equal to the radius
• B. equal to the diameter
• C. equal to the circumference
• D. one metre

Explanation: Since the arc length equals the radius times the angle in radians, an arc of one radius length subtends exactly one radian, about 57.3 degrees. A full circle is therefore 2 pi radians, because the circumference is 2 pi times the radius. Defining the angle this way is what makes the arc length formula and the link between linear and angular quantities so simple.

2. An angle of 180 degrees expressed in radians is

• A. pi over 2
• B. pi
• C. 2 pi
• D. 3 pi over 2

Explanation: A complete revolution of 360 degrees is 2 pi radians, so half a revolution is pi radians, roughly 3.14. A quarter turn of 90 degrees is pi over 2. Converting between the two is simply multiplying by pi over 180 in one direction and by 180 over pi in the other.

3. The relationship between linear speed v and angular speed omega for a body moving in a circle of radius r is

• A. v equals omega divided by r
• B. v equals r divided by omega
• C. v equals r omega
• D. v equals r squared omega

Explanation: In one second the body sweeps an angle omega, covering an arc of length r omega, so the linear speed is r omega with omega in radians per second. This is why two points on a rotating disc share the same angular speed while the outer point moves faster in a straight line sense. The same relationship links angular and linear acceleration as a equals r alpha.

4. A wheel rotating at 300 revolutions per minute has an angular speed of about

• A. 5 radians per second
• B. 31.4 radians per second
• C. 300 radians per second
• D. 1885 radians per second

Explanation: 300 revolutions per minute is 5 revolutions per second, and each revolution is 2 pi radians, so omega is 5 times 2 pi, which is 31.4 radians per second. Forgetting to convert revolutions into radians leaves the answer as 5, the commonest error here. The value 1885 comes from using minutes rather than seconds.

5. A body moving in a circle at constant speed

• A. has zero acceleration
• B. is accelerating, because the direction of its velocity is continuously changing
• C. has a constant velocity
• D. experiences no net force

Explanation: Velocity is a vector, so changing its direction is a change in velocity even when the magnitude is fixed, and that change per unit time is the centripetal acceleration directed towards the centre. A net force must therefore act, supplied by tension, gravity, friction or the normal force depending on the situation. This is the single most tested idea in circular motion.

6. The centripetal acceleration of a body moving in a circle of radius r with speed v is

• A. v squared divided by r, directed towards the centre
• B. v squared times r, directed away from the centre
• C. v divided by r squared, directed along the tangent
• D. zero, since the speed is constant

Explanation: The acceleration has magnitude v squared over r, equivalently r omega squared, and always points at the centre, which is why it is called centripetal or centre seeking. It changes only the direction of motion and never the speed, since it is perpendicular to the velocity at every instant. That perpendicularity is also why it does no work.

7. A car of mass 1000 kg takes a bend of radius 50 m at 10 m per second. The centripetal force needed is

• A. 200 N
• B. 2000 N
• C. 5000 N
• D. 10000 N

Explanation: The force is mv squared over r, which is 1000 times 100 divided by 50, giving 2000 N. On a level road this force must come from friction between the tyres and the surface, so if the required force exceeds the maximum friction available the car skids outwards. Doubling the speed would quadruple the force, which is why bends are so much more dangerous at speed.

8. The so-called centrifugal force experienced by a passenger on a turning bus is

• A. a real force acting outwards on the passenger
• B. the reaction to the passenger's weight
• C. an apparent force arising because the passenger is in a rotating, non inertial frame
• D. always larger than the centripetal force

Explanation: In the ground frame the passenger simply continues in a straight line by inertia while the bus turns beneath them, so no outward force is needed to explain the motion. The outward push is felt only because the observer is accelerating with the bus, which makes it a fictitious or pseudo force. The genuine force in the situation is the inward one supplied by the seat and the floor.

9. For a car to take a banked curve safely without relying on friction, the correct banking angle depends on

• A. the mass of the car only
• B. the speed and the radius of the curve
• C. the weight of the passengers
• D. the width of the road

Explanation: Setting the horizontal component of the normal force equal to mv squared over r gives tan theta equal to v squared over rg, and the mass cancels completely. That is why a banked track works equally well for a light car and a loaded truck at the same design speed. The angle is correct for one speed only, with friction covering the difference at others.

10. The time period T and the angular speed omega of circular motion are related by

• A. T equals omega over 2 pi
• B. T equals 2 pi omega
• C. T equals 2 pi over omega
• D. T equals omega squared

Explanation: One complete revolution covers 2 pi radians, so the time taken is 2 pi divided by the angular speed. A larger omega therefore means a shorter period, which is the inverse relationship the formula expresses. Frequency, being the reciprocal of the period, is omega divided by 2 pi.