Resolving forces

A particle moves under the actions of two forces, F₁ and F₂

F₁ has magnitude 17 newtons and acts due East. F₂ has magnitude 26 newtons and acts at a bearing of 310°

The resultant of F₁ and F₂ is R

Show that the magnitude of R is 17.0 newtons, correct to three significant figures.

Find the angle that R makes with F₁

Give your answer to the nearest degree.

A third force, F₃, acts upon the particle so that the particle is in equilibrium.

State the magnitude of F₃

State the bearing on which F₃ acts.

– 2 16 In this question use g = 9.8 m s

A sledge is pulled in a straight line up a rough path, as shown in the diagram.

The path is inclined at an angle of 20° to the horizontal.

The sledge is pulled by a light, inextensible rope inclined at an angle of 30° to the path.

The mass of the sledge is 10 kilograms.

In one model, the sledge moves at a constant speed and experiences a combined resistance force of 15 newtons.

Find the tension in the rope for this model.

In a different model, the sledge experiences no air resistance.

The tension in the rope for this model is 54 N

The coefficient of friction between the sledge and the path is 0.2

Find the acceleration of the sledge for this model.

State an assumption you have used to answer both parts (a) and (b).

A wooden toy comprises a train engine and a trailer connected to each other by a light, inextensible rod.

The train engine has a mass of 1.5 kilograms. The trailer has a mass 0.7 kilograms.

A string inclined at an angle of 40 above the horizontal is attached to the front of the train engine.

The tension in the string is 2 newtons.

As a result the toy moves forward, from rest, in a straight line along a horizontal surface with acceleration 0.06 m s² as shown in the diagram below.

As it moves the train engine experiences a total resistance force of 0.8 N

Show that the total resistance force experienced by the trailer is approximately 0.6 N

At the instant that the toy reaches a speed of 0.5 m s¹ the string breaks.

As a result of this the train engine and trailer decelerate at a constant rate until they come to rest, having travelled a distance of h metres.

It can be assumed that the resistance forces remain unchanged.

Find the tension in the rod after the string has broken.

Find h

State one modelling assumption that you have used about the rod when answering part (b)(i).

An object, O, of mass m kilograms is hanging from a ceiling by two light, inelastic strings of different lengths.

The shorter string, of length 0.8 metres, is fixed to the ceiling at A.

The longer string, of length 1.2 metres, is fixed to the ceiling at B.

This object hangs 0.6 metres directly below the ceiling as shown in the diagram.

Show that the tension in the shorter string is over 30% more than the tension in the longer string.

The tension in the longer string is known to be 2 g newtons.

Find the value of m.

In this question use 9.8 m s²

A rough wooden ramp is 10 metres long and is inclined at an angle of 25 above the horizontal. The bottom of the ramp is at the point O.

A crate of mass 20 kg is at rest at the point A on the ramp.

The crate is pulled up the ramp using a rope attached to the crate.

Once in motion, the rope remains taut and parallel to the line of greatest slope of the

The tension in the rope is 230 N

The crate accelerates up the ramp at 1.2 m s²

Find the coefficient of friction between the crate and the ramp.

The crate takes 3.8 seconds to reach the top of the ramp.

Find the distance OA.

Other than air resistance, state one assumption you have made about the crate in answering part (b)(i).

6 11 A number of forces act on a particle such that the resultant force is N

Calculate the total of the other forces acting on the particle.

Circle your answer.

Block A, of mass 0.2 kg, lies at rest on a rough plane. The plane is inclined at an angle to the horizontal, such that tan

A light inextensible string is attached to A and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope.

The other end of this string is attached to particle B, of mass 2 kg, which is held at rest so that the string is taut, as shown in the diagram below.

B is released from rest so that it begins to move vertically downwards with an

Show that the coefficient of friction between A and the surface of the inclined plane is 0.17

In this question use 9.81 m s²

When A reaches a speed of 0.5 m s¹ the string breaks.

Find the distance travelled by A after the string breaks until first coming to rest.

State an assumption that could affect the validity of your answer to part (b)(i).

Lizzie is sat securely on a wooden sledge.

The combined mass of Lizzie and the sledge is M kilograms.

The sledge is being pulled forward in a straight line along a horizontal surface by means of a light inextensible rope, which is attached to the front of the sledge.

This rope stays inclined at an acute angle above the horizontal and remains taut as the sledge moves forward.

The sledge remains in contact with the surface throughout.

The coefficient of friction between the sledge and the surface is and there are no other resistance forces.

Lizzie and the sledge move forward with constant acceleration, a m s²

The tension in the rope is a constant T Newtons.

Show that

It is known that when M 30, 30, and T 40, the sledge remains at rest.

Lizzie uses these values with the relationship formed in part (a) to find the value for

Explain why her value for may be incorrect.