Circular Motion
1 An Earth satellite is required to be in a circular orbit at a distance of 7.5 x 106 m from the centre of the Earth. The central force is due to the gravitational force. The acceleration due to the Earth’s gravity at this point is 7.0 ms-2.
Find:
(a) the required satellite speed
(b) the period of revolution of the satellite.
2 What would be the period of rotation of the Earth about its axis if its speed of rotation increased to such an extent that an object at the equator became ‘weightless’?
(Hint: equate mg to mv²/r).
3 A sphere of mass 0.20 kg is rotating in a circular path at the end of a string 0.80 m long. The other end of the string is fixed. The period of the motion is 0.25 s.
(a) Calculate the tension in the string, which you may assume to be horizontal.
(b) In practice the string is not horizontal. Explain why this is so.
(c) Draw a force diagram for the sphere. From this calculate the angle the string would make with the horizontal.
4 The moon takes 27.3 days (2.0 x 106 s) to complete one orbit of the Earth.
The distance between the centres of the Earth and Moon is 4.0 x 108 m.
Calculate the magnitude of the Moon’s acceleration towards the Earth.
5 A ball of mass 2.0 kg is attached to a string 1.6 m long and is made to travel in a vertical circle. The ball passes its highest point with a speed of 5.0 ms-1.
(a) What is the kinetic energy of the ball at its highest point?
(b) What is its potential energy when it is at the highest point (with reference to its lowest point)?
(c) What is its kinetic energy at the lowest point?
(d) What is its speed at the lowest point?
(e) What is the tension in the string at the highest and lowest points?
(f) What is the least speed the ball could have at the highest point in order to be able to complete a vertical circle at all?
6 An old humpback bridge has a radius of curvature of 20 m. What is the maximum speed at which a car can pass over this bridge if the car is not to leave the road surface?
7 (a) A pail of water is swinging in a vertical circle of radius 1.2 m, so that the water does not fall out. What is the minimum linear speed required for the pail of water.
(b) Convert this speed into an angular velocity.
8 An object of mass 0.20 kg is connected by a string to an object of half its mass. The smaller mass is rotating at a radius of 0.15 m on a table which has a frictionless surface. The larger mass is suspended through a hole in the middle of the table.
Calculate the number of revolutions per minute the smaller mass must make so that the larger mass is stationary.
Banking of a Track
9 A circular track of radius 60 m is banked at angle. A car is driven round the track at 20 ms-1.
(a) Draw a diagram showing the forces acting on the car.
(b) Calculate the angle of banking required so that the car can travel round the track without relying on frictional forces (i.e. no side thrust supplied by friction on the track surface).
Conical Pendulum
10 A small object of mass m revolves in a horizontal circle at constant speed at the end of a string of length 1.2 m. As the object revolves, the string sweeps out the surface of a right circular cone.
The cone has semi-angle 30°.
Calculate:
(a) the period of the motion;
(b) the speed of the object.
[Hint: try resolving the tension in the string into horizontal and vertical components.]
