Free Angular and Linear Quantities MCQs with Answers

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.