Further Forces & Newton's Laws (CIE AS Maths: Mechanics)

The following force diagram shows three forces acting on a particle:

Find the magnitude and direction of the resultant forces acting on the particle in the horizontal and vertical directions.

The following force diagram shows three forces acting on a particle:

Show that the resultant force on the particle in the horizontal direction is zero.

Given that the particle is in equilibrium, find the exact value of .

A particle of mass  is suspended by a light inextensible string, with the other end of the string attached to a fixed point .  With the string at an angle of  to the vertical, equilibrium is maintained by a horizontal force of 12 N which acts on the particle as shown in the diagram below:

Find

the value of .

A particle of mass 10 kg is sliding down a smooth slope that is angled at  to the horizontal.

Calculate the acceleration of the particle down the slope.

Calculate the normal reaction force of the slope on the particle.

A particle of mass 7 kg is being pulled along a rough horizontal plane by a force of 45 N acting at an angle of  to the horizontal.

The coefficient of friction between the particle and the plane is 0.4, and the particle is moving to the right under the influence of the force.

By first finding the normal reaction force, calculate the frictional force between the particle and the plane and state its direction.

Hence calculate the acceleration of the particle.

A particle of mass 0.7 kg is on a rough plane which is angled at  to the horizontal.  The particle is being held at rest by a force of  N acting up the plane along the line of greatest slope.

Given that the coefficient of friction between the particle and the plane is 0.35, and that the particle is on the point of slipping up the plane, find the value of .

A particle of mass 3 kg is released from rest on a rough plane which is inclined at  to the horizontal.  The coefficient of friction between the particle and the plane is 0.2.

Determine the distance the particle will slide down the plane in the first 3 seconds after it is released.

A particle of mass 3 kg is projected up a rough plane which is inclined at an angle of  to the horizontal.  It is projected up the line of greatest slope with an initial velocity of , and it comes to instantaneous rest after moving a distance of 1.91 m up the slope.  The coefficient of friction between the particle and the slope is .

Show that while the particle is moving up the plane it experiences a deceleration of ,  where  is the constant of acceleration due to gravity.

Hence find the value of  .

A box  of mass 5 kg rests on a rough horizontal table.  It is connected by a light inextensible string to a metal sphere  of mass 2 kg.  The string passes over a smooth light fixed pulley at the edge of the table so that  is hanging vertically downwards as shown in the diagram below:

The string between  and the pulley is horizontal, and the coefficient of friction between  and the table is 0.35.

The system is released from rest with the string taut.

Calculate

as  descends.

Two particles  and , of masses 4 kg and 3 kg respectively, are connected by means of a light inextensible string.  Particle  is held motionless on a rough fixed plane inclined at  to the horizontal.  The string passes over a smooth light pulley fixed at the top of the plane so that  is hanging vertically downwards as shown in the diagram below:

The string between  and the pulley lies along a line of greatest slope of the plane, and  hangs freely from the pulley.  The coefficient of friction between particle  and the plane is .

The system is released from rest with the string taut.

Calculate the acceleration of the two objects and the tension in the string as  descends.

After descending for 3.2 seconds, particle  strikes the ground and immediately comes to rest.  Particle  continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle  between the time that the system is first released from rest and the time that particle  comes momentarily to rest again after  has struck the ground.

The following force diagram shows three forces acting on a particle:

Find the magnitude and direction of the resultant forces acting on the particle in the horizontal and vertical directions.

The following force diagram shows three forces acting on a particle:

Given that the particle is in equilibrium, find:

the angle, , that F makes with the horizontal.

Give both your answers correct to 3 significant figures.

A particle of mass m kg  hangs in equilibrium, suspended by two light inextensible strings. The strings are inclined at 25° and 70° to the horizontal, as shown in the diagram below:

Given that the tension in the string angled at 70° to the horizontal is 56 N, find the value of m.

A particle of mass 12 kg is being pushed up a smooth slope by a force of 50 N that acts horizontally. The slope is inclined at  to the horizontal, as shown in the diagram below:

Calculate the acceleration of the particle up the slope.

Calculate the normal reaction force of the slope on the particle.

A particle of mass 15 kg is sliding down a rough slope that is angled 25° to the horizontal. The coefficient of friction between the particle and the slope is 0.3.

Calculate the frictional force between the particle and the slope.

Hence calculate the acceleration of the particle down the slope.

A particle of mass 0.9 kg is at rest on a rough horizontal plane.  A force of magnitude P N is acting on the particle at an angle of 40° to the horizontal.

Given that the coefficient of friction between the plane and the particle is 0.3, and that the particle is on the point of moving to the right under the influence of the force, find the value of P.

A particle of mass m kg is released from rest on a rough plane inclined at θ° to the horizontal, where .  The coefficient of friction between the particle and the plane is .

Given that the particle remains motionless after it is released, show that .

A particle of mass 2 kg is projected up a rough plane which is inclined at an angle of 20° to the horizontal.  It is projected up the line of greatest slope with an initial velocity of , and it comes to instantaneous rest after moving a distance of 4.85 m up the slope.  The coefficient of friction between the particle and the slope is 0.2.

Find the value of .

Two particles A and B, of masses 2.7 kg and 2.2 kg respectively, are connected by means of a light inextensible string.  Particle A is held motionless on a rough fixed plane inclined at 25° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the plane so that B is hanging vertically downwards as shown in the diagram below:

The string between A and the pulley lies along a line of greatest slope of the plane, and B hangs freely from the pulley.  The coefficient of friction between particle A and the plane is μ.

The system is released from rest with the string taut.  Given that particle B descends 1.82 m in the first 3 seconds after it is released, find the value of μ.

Two particles A and B, of identical mass, are connected by means of a light inextensible string. Particle A is held motionless on a rough fixed plane inclined at 30° to the horizontal, and that plane is connected at its top to another rough fixed plane inclined at 70° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the two planes so that B is hanging downwards in contact with the second plane.   This situation is shown in the diagram below:

The parts of the string between A and the pulley and between B and the pulley each lie along a line of greatest slope of the respective planes. The coefficient of friction between the particles and the planes is 0.15 in both cases.

The system is released from rest with the string taut, and with particle B a vertical distance of from the ground.  Particle B descends along the slope until it reaches the ground, at which point it immediately comes to rest.  Particle A continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle A between the time that the system is first released from rest and the time that particle A comes momentarily to rest again after B has reached the ground.

The following force diagram shows three forces acting on a particle:

Given that the resultant force on the particle in the vertical direction is 44.8 N downwards, find the size of the angle .  Give your answer in degrees correct to 3 significant figures.

In a warehouse, a lifting device is used to lift a load of weight  by means of two lifting cables as shown in the diagram below:

The tension, ,  is the same in each of the lifting cables,  and the angle that each of the cables makes with the vertical, ,  is also the same.  For purposes of this question any influences on the motion of the load other than the weight of the load and the tension in the two cables may be ignored.

Given that the load is initially motionless, find an inequality in terms of  and  that the weight  must satisfy if it is to be lifted by the cables.

Given that the load must still be accelerating in the upwards direction when ,  find an inequality in terms of  that the tension  must satisfy.

A smooth bead  with a mass of  grams has been threaded onto a light inextensible string.  The ends of the string are fixed to two points  and , where  is vertically above .  The bead is held in equilibrium by a horizontal force of 3 N, with sections   and  of the string making angles of  and  respectively with the vertical as shown in the diagram below:

Find the value of .

A particle is being pulled up a smooth slope by a force of 60 N that acts at an angle of  to the slope.  The slope is inclined at  to the horizontal, as shown in the diagram below:

The particle experiences an acceleration of  parallel to the slope in the upwards direction.

Use the information provided above to calculate the mass of the particle.

Calculate the normal reaction force of the slope on the particle.

A laboratory apparatus contains a rough inclined plane that may be inclined at different angles to the horizontal.  When the plane is at an angle of  to the horizontal, where ,  a small mass is released from rest on the plane and remains motionless.  After the angle is increased to ,  where ,  the same small mass is released from rest on the plane and begins to slide down the slope of the plane.

Show that

where  is the coefficient of friction between the small mass and the plane.

A particle of mass 0.5 kg is at rest on a rough plane which is inclined at  to the horizontal.  The particle is being acted upon by a force of 6 N, directed at an angle of  to the plane.  The line of action of the force is in the same vertical plane as the line of greatest slope of the inclined plane.

Given that the coefficient of friction between the particle and the plane is 0.4, and that the particle is on the point of slipping up the plane, find the value of .

You may use the following trigonometric identity in your working without proof:

A particle of mass 0.8 kg is released from rest on a rough slope that is angled at  to the horizontal, where .  After 4 seconds the speed of the particle is .  Find the value of the coefficient of friction  between the particle and the slope.

A particle of mass  kg is projected up a rough plane which is inclined at an angle of  to the horizontal.  It is projected up the line of greatest slope with an initial velocity of  metres per second, and it comes to instantaneous rest in  seconds after moving a distance of metres up the slope. The coefficient of friction between the particle and the slope is .

Show that:

After coming to instantaneous rest, the particle begins to slide back down the slope, and after t₂ seconds it has returned to its starting point.

Find an expression for t₂ in terms of .

You have landed on the planet Hephaestia, where the gravitational constant of acceleration g has a different value than it does on Earth. 

Your spaceship contains a device which may be used to find the value of g on any planet.  In this device a particle A with mass m_A kg is connected by a light inextensible string to a light scale-pan.  A force meter C with mass m_C kg is placed in the scale-pan, and a small block B with mass m_B kg is placed on top of C. A is held in place on a rough plane angled at θ° to the horizontal, where the coefficient of friction between A and the plane is µ.  The string passes over a smooth light pulley fixed at the top of the plane so that the scale-pan is hanging vertically downwards as shown in the diagram below:

With the string between A and the pulley lying along a line of greatest slope of the plane, A is projected down the plane with a velocity of parallel to the string.  After a time of t seconds the system comes momentarily to rest, and then the scale-pan begins to descend under the force of gravity, pulling mass A up the slope behind it.

When the scale-pan is initially moving upwards, the force exerted by B on C is denoted by F_BC↑. When the scale-pan begins to descend, the force exerted by B on C is denoted by F_BC↓. The force meter is only able to record the difference ΔF between these two values, where

Show that

For your ship’s device, 

Find the value of g on Hephaestia, given that the value recorded for ΔF is 1.286 N.