Motion in a vertical circle

One end of a light inextensible string of length r m is attached to a fixed point O. The other end is attached to a particle P of mass m kg.

Particle P hangs in equilibrium vertically below O with the string taut. An initial horizontal impulse is applied to P so that it instantaneously starts moving horizontally with a speed of u m s^-1.

Particle P subsequently moves in complete circles in a vertical plane containing O.

When the line OP makes an angle of θ with the downward vertical through O, the speed of P is v m s^-1 and the tension in the string is TN.

Show that T = u² + α gr + β gr cos θ) where α and β are constants to be determined.

Determine, in terms of m, g and r, the minimum initial impulse required to ensure that P

describes complete vertical circles around O with the string remaining taut.

Find the tangential acceleration of P at A, stating its direction.

Determine the radial acceleration of P at A, stating its direction.

Find the magnitude of the force in the rod when P is at A, stating whether the rod is in tension

or compression.

The motion is now stopped when P is at A, and P is then projected in such a way that it now describes horizontal circles at a constant speed with θ = 40 ° (see diagram).

Find the speed of P.

Explain why, wherever P ’s motion is initiated from and whatever its initial velocity, it is not

possible for P to describe horizontal circles at constant speed with θ = 9 0 °.

Show that at this instant the speed of P is 4.5 ms^-1.

Find the magnitude and direction of the radial acceleration of P at this instant.

Find the magnitude of the tangential acceleration of P at this instant.

The training rig for a parachutist comprises a fixed platform and a fixed hook, H. The platform is 3.5 m above horizontal ground level. The hook, which is not directly above the platform, is 6.5 m above the ground.

One end of a light inextensible cord of length 4.5 m is attached to H and the other is attached to a trainee parachutist of mass 90 kg standing on the edge of the platform with the cord straight and taut.

The trainee is then projected off the platform with a velocity of 7 m s^-1 perpendicular to the cord in a downward direction. The motion of the trainee all takes place in a single vertical plane and while the cord is attached to H it remains straight and taut.

When the speed of the trainee reaches 5.5 m s^-1 the cord is detached from H and the trainee then moves under the influence of gravity alone until landing on the ground (see diagram).

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

Show that at the instant before the cord is detached from H, the tension in the cord has a

magnitude of 1005.5 N.

The point on the ground vertically below the edge of the platform is denoted by O. The point on the ground where the trainee lands is denoted by T.

Determine the distance OT.

The ground around T is in fact an elastic mat of thickness 0.5 m which is angled so that it is perpendicular to the direction of motion of the trainee on landing. The mat, which is very rough, is modelled as an elastic spring of natural length 0.5 m. It is assumed that the trainee strikes the mat at ground level and is brought to rest once the mat has been compressed by 0.3 m.

Determine the modulus of elasticity of the mat. Give your answer to the nearest integer.

Show that v² = 50.7 - 14.7 cos θ.

Given that the string breaks when the tension in it reaches 200 N, find the angle that OA turns

through between the instant that A is projected and the instant that the string breaks.