Angular speed

A particle P of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m. The other end of the string is attached to the fixed point A. A second light inextensible string of length 0.6 m has one end attached to P and the other end attached to the fixed point B. The point B is vertically below A such that AB = 0.8 m, as shown in Figure 4.

The particle moves in a horizontal circle with constant angular speed ω radians per second, with both strings taut.

Given that the tension in the string from P to A is twice the tension in the string from P to B, find the value of ω.

Figure 2 shows a hemispherical bowl of internal radius 10d that is fixed with its circular rim horizontal.

The centre of the circular rim is at the point O.

A particle P moves with constant angular speed on the smooth inner surface of the bowl.

The particle P moves in a horizontal circle with radius 8d and centre C.

Find, in terms of g, the exact magnitude of the acceleration of P.

The time for P to complete one revolution is T.

Find T in terms of d and g.

A hollow right circular cone, of internal base radius 0.6 m and height 0.8 m, is fixed with its axis vertical and its vertex V pointing downwards, as shown in Figure 4.

A particle P of mass m kg moves in a horizontal circle of radius 0.5 m on the rough inner surface of the cone.

The particle P moves with constant angular speed ω rads¹

The coefficient of friction between the particle P and the inner surface of the cone is 0.25

Find the greatest possible value of ω

A small smooth ring R of mass m is threaded onto a light inextensible string. One end of the string is attached to a fixed point A and the other end of the string is attached to the fixed point B such that B is vertically above A and AB = 6 a

The ring moves with constant angular speed ω in a horizontal circle with centre A. The string is taut and BR makes a constant angle θ with the downward vertical, as shown in Figure 2.

The ring is modelled as a particle.

find, in terms of m and g, the magnitude of the tension in the string,

find ω in terms of a and g

A particle P of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm. The other end of the string is attached to a fixed point A that is vertically above the point O on a smooth horizontal table, such that OA = 40 cm. The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre O, as shown in Figure 4.

The particle is moving with a constant angular speed of 3 radians per second.

(a) Find (i) the tension in the string,

the normal reaction between P and the table.

The angular speed of P is now gradually increased.

Find the angular speed of P at the instant P loses contact with the table.