Motion in a vertical circle

In this question use g = 10 m s^–2

A particle P of mass 0.4 kg is attached to one end of a light inextensible string of length 1.5 m. The other end of the string is attached to a fixed point O. The particle is at rest vertically below O with the string taut.

The particle is then projected horizontally with speed 6 m s^–1

When the string has turned through an angle θ, the string is still taut and the speed of P is vm s^–1, as shown in Figure 8.

A smooth solid hemisphere has radius r and the centre of its plane face is O. The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6. A small stone is at the point A, the highest point on the surface of the hemisphere. The stone is projected horizontally from A with speed U. The stone is still in contact with the hemisphere at the point B, where OB makes an angle θ with the upward vertical. The speed of the stone at the instant it reaches B is v.

The stone is modelled as a particle P and air resistance is modelled as being negligible.

Use the model to find v² in terms of U, r, g and θ

When P leaves the surface of the hemisphere, the speed of P is W.

After leaving the surface of the hemisphere, P moves freely under gravity until it hits the ground.

Find the speed of P as it hits the ground, giving your answer in terms of r and g.

At the instant when P hits the ground it is travelling at α to the horizontal.

Find the value of α.

A smooth hemisphere of radius a is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is O.

A particle P of mass M is disturbed from rest at the highest point of the hemisphere.

When P is still on the surface of the hemisphere and the radius from O to P is at an angle θ to the vertical,

• the speed of P is v

• the normal reaction between the hemisphere and the particle is R, as shown in Figure 2.

Find, in terms of a and g, the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

end of the string is attached to a fixed point O. The package hangs at rest vertically below O with the string taut and is then projected horizontally with speed u, as shown in Figure 5.

When OP has turned through an angle θ and the string is still taut, the tension in the string is T

The package is modelled as a particle and the string as being light and inextensible.

Given that P moves in a complete vertical circle with centre O

find, in terms of a and g, the minimum possible value of u

find, in terms of g, the magnitude of the acceleration of P at the instant when OP is horizontal.

Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.

A light inextensible string of length a has one end attached to a fixed point O. The other end of the string is attached to a small stone of mass m. The stone is held with the string taut and horizontal. The stone is then projected vertically upwards with speed U.

The stone is modelled as a particle and air resistance is modelled as being negligible.

Assuming that the string does not break, use the model to

find the least value of U so that the stone will move in complete vertical circles.

The string will break if the tension in it is equal to

find the total angle that the string has turned through, from when the stone is projected vertically upwards, to when the string breaks,

find the magnitude of the acceleration of the stone at the instant just before the string breaks.

A particle P of mass m is attached to one end of a light inextensible string of length l. The other end of the string is attached to a fixed point O. The particle is held with the string taut and OP horizontal. The particle is then projected vertically downwards with speed u, where u²= gl. When OP has turned through an angle α and the string is still taut, the speed of P is v, as shown in Figure 5. At this instant the tension in the string is T.

Find, in terms of g and l, the speed of P at the instant when the string goes slack.

Find, in terms of l, the greatest vertical height reached by P above the level of O.