Resolving forces

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 block of mass m kg is on smooth horizontal ground with one end of a light inextensible rope attached to its upper surface. The other end of the rope is attached to an object of mass 5 kg. The rope passes over a small smooth pulley, and the object hangs vertically below the pulley. The part of the rope between the block and the pulley makes an angle of 50° with the horizontal. A force of magnitude X N acts on the block at an angle of 20° above the horizontal in the vertical plane containing the rope (see diagram).

You are given that the block is in equilibrium.

Determine the value of X.

You are also given that the magnitude of the contact force exerted by the ground on the block is 147 N.

Determine the value of m.

The points A and B are the lower and upper ends, respectively, of a line of greatest slope on a plane inclined at an angle θ to the horizontal, where sin θ = 0.6 and AB = 1.375 m (see diagram).

A particle P is projected up the plane with speed 6 ms^-1 from A towards B.

The plane at A is fixed to the ground which is horizontal.

The surface of the plane is rough and the coefficient of friction between P and the plane is 0.25.

Show that the speed of P at B is 3.8 ms^-1.

The particle leaves the slope at B and moves freely under gravity.

The particle first lands at a point C on the horizontal ground. The time taken for P to travel from A to C is T seconds.

Determine the value of T.

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.

One end of a light inextensible string is attached to a particle A of mass 2 kg. The other end of the string is attached to a second particle B of mass 3 kg. Particle A is in contact with a smooth plane inclined at 30° to the horizontal and particle B is in contact with a rough horizontal plane.

A second light inextensible string is attached to B. The other end of this second string is attached to a third particle C of mass 4 kg. Particle C is in contact with a smooth plane Π inclined at an angle of 60° to the horizontal.

Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from A to the pulley, and from C to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram).

The coefficient of friction between B and the horizontal plane is μ. The system is released from rest and in the subsequent motion C moves down Π with acceleration a m s^-2.

By considering an equation involving μ, a and g show that a < (2 - 1).

Given that a = , determine the magnitude of the contact force between B and the horizontal

plane. Give your answer correct to 3 significant figures.

One end of a light inextensible string is attached to a particle A of mass 2 kg. The other end of the string is attached to a second particle B of mass 2.5 kg. Particle A is in contact with a rough plane inclined at θ to the horizontal, where cos θ = . The string is taut and passes over a small smooth pulley P at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. Particle B hangs freely below P at a distance 1.5 m above horizontal ground, as shown in the diagram.

The coefficient of friction between A and the plane is μ. The system is released from rest and in the subsequent motion B hits the ground before A reaches P. The speed of B at the instant that it hits the ground is 1.2 ms^-1.

For the motion before B hits the ground, show that the acceleration of B is 0.48 ms^-2.

For the motion before B hits the ground, show that the tension in the string is 23.3 N.

Determine the value of μ.

After B hits the ground, A continues to travel up the plane before coming to instantaneous rest before it reaches P.

Determine the distance that A travels from the instant that B hits the ground until A comes to

instantaneous rest.

The diagram shows a small block B, of mass 0.2 kg, and a particle P, of mass 0.5 kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane.

The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of θ with the horizontal where tan θ =

The system is released from rest. In the first 0.4 seconds of the motion P moves 0.3 m down the plane and B does not reach the pulley.

Find the tension in the string during the first 0.4 seconds of the motion.

Calculate the coefficient of friction between B and the horizontal surface.

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.

One end of a light inextensible string is attached to a particle A of mass m kg. The other end of the string is attached to a second particle B of mass λ m kg, where λ is a constant. Particle A is in contact with a rough plane inclined at 30° to the horizontal. The string is taut and passes over a small smooth pulley P at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. The particle B hangs freely below P (see diagram).

The coefficient of friction between A and the plane is μ.

It is given that A is on the point of moving down the plane.

Find the exact value of μ when λ =