Free Rotational and Circular Motion MCQs with Answers

11. The moment of inertia of a body depends on

• A. its mass only
• B. its mass and how that mass is distributed about the axis of rotation
• C. its angular speed
• D. the torque applied to it

Explanation: Moment of inertia is the sum of each mass element multiplied by the square of its distance from the axis, so moving mass further out increases it sharply even when the total mass is unchanged. This is why a hollow cylinder has a larger moment of inertia than a solid one of the same mass. It plays the same role in rotation that mass plays in linear motion.

12. A skater spinning with arms outstretched pulls the arms inwards. The result is that

• A. the angular speed increases, because the moment of inertia decreases and angular momentum is conserved
• B. the angular speed decreases
• C. the angular momentum increases
• D. nothing changes

Explanation: With no external torque the angular momentum, the product of moment of inertia and angular speed, stays constant, so reducing the moment of inertia must raise the angular speed. The skater's rotational kinetic energy actually increases, and the extra energy comes from the muscular work done pulling the arms in against the outward tendency. The same principle explains the spin of a collapsing star.

13. The angular momentum of a rotating rigid body is given by

• A. I alpha
• B. I omega
• C. m v r squared
• D. half I omega squared

Explanation: Angular momentum is the moment of inertia multiplied by the angular speed, the rotational analogue of linear momentum mv. The product I alpha is torque, the analogue of force, and half I omega squared is rotational kinetic energy. Keeping these three analogues apart is what most rotational questions actually test.

14. A satellite in a circular orbit around the Earth is held in orbit by

• A. the vacuum of space
• B. its own centrifugal force
• C. the gravitational pull of the Earth acting as the centripetal force
• D. the thrust of its engines

Explanation: Gravity supplies exactly the inward force needed to bend the satellite's straight line motion into a circle, so no engine is required once the orbit is established. Setting the gravitational force equal to mv squared over r gives the orbital speed, which depends on the radius but not on the satellite's mass. Astronauts feel weightless because they and the craft are falling towards the Earth together, not because gravity is absent.

15. A geostationary satellite must have an orbital period of

• A. 1 hour
• B. 12 hours
• C. 24 hours
• D. 365 days

Explanation: To stay above the same point on the equator the satellite must complete one orbit in exactly the time the Earth takes to rotate once, which is 24 hours. This fixes the orbital radius at about 42,300 km from the centre of the Earth. Communication and television satellites use this orbit so that a dish on the ground can be aimed once and left fixed.

16. The rotational kinetic energy of a body is

• A. half m v squared
• B. I omega
• C. half I omega squared
• D. I alpha

Explanation: Replacing mass with moment of inertia and linear speed with angular speed in the familiar expression gives half I omega squared. A rolling body has both this and the translational half m v squared, which is why a ball rolling down a slope reaches the bottom more slowly than one that slides without friction. The share of the energy going into rotation depends on the shape of the body.

17. A stone is whirled in a horizontal circle on a string. If the string suddenly breaks, the stone will

• A. fly radially outwards from the centre
• B. move along the tangent to the circle at the point of release
• C. fall straight down at once
• D. spiral outwards while slowing down

Explanation: With the centripetal force gone, no force acts horizontally and Newton's first law takes over, so the stone keeps the velocity it had at that instant, which is directed along the tangent. Gravity then curves the path downwards into a projectile trajectory. The belief that it flies straight outwards comes from thinking of centrifugal force as real.

18. At the top of a vertical circular loop, the minimum speed for a body to maintain contact with the track is given by

• A. v equals the square root of rg
• B. v equals rg
• C. v equals zero
• D. v equals the square root of 2rg

Explanation: At the critical speed the normal reaction falls to zero and gravity alone provides the centripetal force, so mg equals mv squared over r and v is the square root of rg. Any slower and the body leaves the track before reaching the top. This is the calculation behind loop the loop rides and the bucket of water swung overhead.

19. Angular displacement is measured in

• A. metres
• B. radians
• C. metres per second
• D. newtons

Explanation: The radian is the SI unit of angle, and because it is defined as a ratio of two lengths it is dimensionless, which is why the formula v equals r omega comes out in metres per second without any conversion factor. Degrees and revolutions are convenient in practice but must be converted to radians before use in these formulae. Linear displacement, by contrast, is measured in metres.

20. Two points lie on a rotating disc, one twice as far from the axis as the other. Compared with the inner point, the outer point has

• A. twice the angular speed and the same linear speed
• B. the same angular speed and twice the linear speed
• C. half the angular speed
• D. the same linear speed

Explanation: Every point on a rigid rotating body sweeps the same angle in the same time, so the angular speed is common to all of them. Linear speed is r omega, so doubling the radius doubles the speed along the arc. This is why the rim of a grinding wheel moves so much faster than a point near its centre.