# Archive | M3

## Sphere Attached to a String Resting Against a Smooth Wall

When an object is in contact with a wall, it will experience a normal reaction force. If the object is held up by an inelastic string making an anglewith the downwards vertical, the tension in the string will have both vertical and horizontal components. If the system is in equilibrium, the sum of all the […]

## Particle on Elastic String Executing Horizontal

A particle suspended from an elastic string will extend the string slightly and if the particle is made to move in a horizontal circle, the will be a component of the centripetal force extending the string a little more. To find the extension and the tension in the string, we can resolve vertically and horizontally. […]

## Motion of a Particle on Surface of Smooth Sphere

A particle moving on the surface of a smooth sphere will remain in contact with the sphere as long as the reaction force is greater than zero. We can use this as a condition to find the point at which the particle loses contact with the sphere. If at some point the line joining the […]

## Maximum Angular Speed for Horizontal Circular Motion of a Particle on an Elastic String

A particle moving in a horizontal circle, suspended from the ceiling by an elastic string of natural lengthand modulus of elasticitymust move with an angular velocity less than some minimum Applyinghorizontally gives (1) so (1) can be writtenhence (2) Writeso thatwhereis the extension of the string. Substitute this into (2) to obtain Rearranging this gives […]

## Centre of Mass of a Solid of Revolution

When a curveis rotated about the– axis, the centre of mass of the solid generated will line on the– axis because of the symmetry of the curve. The centre of mass will not lie on the– axis however. If the centre of gravity of a volume of revolution of massis at then taking moments about […]

## Motion of a Particle Attached to Two Springs on a Smooth Surface

A particle between two springs on a table can execute simple harmonic motion if pulled to one side. A particle of mass m is attached to two strings of natural lengthsandand force constants andrespectively. The ends of the springs are attached to endpoints A and B, so that and both springs are initially in tension. […]

## Circular Horizontal Motion of a Ring Threading an Inelastic String

If a ring of mass m is threaded on a string of lengthtied to points A and B with B vertically below A and l>AB the ring can be made to execute circular horizontal motion. Resolving vertically for the ring gives (1) Usinghorizontally gives (2) gives Hence (3) Simple trigonometry givesandand since the length of […]

## Centres of Mass Tables

Body Centre of Mass Solid hemisphere, radius from centre Hemispherical shell, radius from centre Circular arc of lengthradius from centre Sector of circle of arc length from centre Solid right circular cone, height from vertex Conical shell, height From vertex

## Bungee Jumping

When someone attaches themselves to a piece of elastic and jumps off a bridge, energy initially all gravitational potential energy is changed into kinetic energy as the jumper falls and finally into elastic potential energy as the elastic reaches a maximum length. Suppose that the unstretched length of elastic isits modulus of elasticity isand the […]