Equilibrium of a particle

Two horizontal forces of magnitudes 17 N and 12 N act at a point O along bearings of 050° and 340° respectively (see diagram).

Determine the magnitude and bearing of the resultant force.

A third horizontal force F is now applied at O. The three forces are in equilibrium.

State the magnitude of F and give the bearing along which it acts.

A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of 15° with the horizontal (see diagram).

Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N, find

the tension in the rope.

A rectangular block B is at rest on a horizontal surface. A particle P of mass 2.5 kg is placed on the upper surface of B. The particle P is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle Q of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between P and the pulley is horizontal (see diagram).

The particles are released from rest with the string taut. It is given that B remains in equilibrium while P moves on the upper surface of B. The tension in the string while P moves on B is 16.8 N.

Find the acceleration of Q while P and B are in contact.

Determine the coefficient of friction between P and B.

In this question the unit vectors i and j are in the directions east and north respectively.

At time t seconds, where t ⩾ 0, a particle P of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force (- 8 i - 54 j)N and a variable horizontal force (4ti + 6(2t - 1)²j)N.

Determine the value of t when the forces acting on P are in equilibrium.

It is given that P is at rest when t = 0.

Determine the speed of P at the instant when P is moving due north.

Determine the distance between the positions of P when t = 0 and t = 3.

Three forces, of magnitudes 4 N, 6 N and P N, act at a point in the directions shown in the diagram.

The forces are in equilibrium.

Show that θ = 41.4°, correct to 3 significant figures.

Hence find the value of P.

The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.

Find

the magnitude of the resultant of the two remaining forces,

the direction of the resultant of the two remaining forces.