Newton's Second Law (F=ma) (Edexcel A Level Maths: Mechanics): Exam Questions

A car of mass 1200 kg is towing a trailer of mass 400 kg along a straight horizontal road using a tow rope, as shown in Figure 2.

The rope is horizontal and parallel to the direction of motion of the car.

• The resistance to motion of the car is modelled as a constant force of magnitude newtons

• The resistance to motion of the trailer is modelled as a constant force of magnitude newtons

• The rope is modelled as being light and inextensible

• The acceleration of the car is modelled as ms^–2

The driving force of the engine of the car is 7400 N and the tension in the tow rope is 2400 N.

Using the model, find the value of .

In a refined model, the rope is modelled as having mass and the acceleration of the car is found to be ms^–2.

State how the value of compares with the value of .

State one limitation of the model used for the resistance to motion of the car.

A vertical rope has its end attached to the top of a small lift cage.

The lift cage has mass 40 kg and carries a block of mass 10 kg, as shown in Figure 1.

The lift cage is raised vertically by moving the end of the rope vertically upwards with constant acceleration 0.2 ms⁻².

The rope is modelled as being light and inextensible and air resistance is ignored.

Using the model, find the tension in the rope .

Using the model, find the magnitude of the force exerted on the block by the lift cage.

In a show, a performer is lifted up into the air by a cable attached to a harness.  The performer has a mass of 54 kg, and at the start of the lift is standing stationary on the floor. 

The performer accelerates vertically upwards at a constant rate, and after 2 seconds has reached a height of 3 metres.

Find the tension in the cable during the lift.

At the end of the show, the performer is lowered vertically downwards back to the floor. 

During the initial part of the descent the downwards acceleration is constant, and its magnitude is the same as the magnitude of the upwards acceleration was while the performer was being lifted.

Find the tension in the cable during the initial part of the descent.

A child is pulls a cart along a horizontal path using a horizontal rope attached to the front of the cart.

The cart has a total mass of 15 kg. As it moves it is subject to a constant resistive force of magnitude 2 N. 

The cart starts from rest and accelerates at a constant rate. After 5 seconds it has reached a speed of .

Find the tension in the rope.

Two particles, and , are connected by a light inextensible string. Particle hangs directly below particle . 

Particle  has a mass of 5 kg and particle  has a mass of 15 kg.

A force of  300 N  is applied vertically upwards to particle , causing the particles to accelerate.

Find

(i) the initial acceleration of the particles,

(ii) the tension in the string.

A train locomotive of mass 7 000 kg and a carriage of mass 2 000 kg are at rest on a section of horizontal track. 

The connection between the locomotive and the carriage may be modelled as a light rod parallel to the direction of their motion along the track.  

The resistances to motion of the locomotive and the carriage are modelled as constant forces of 2 300 N and 1 000 N respectively.

The locomotive begins to accelerate in the backwards direction, towards the carriage, with its engine providing a constant driving force of 15 000 N.

Find the magnitude of the acceleration of the locomotive and carriage.

Find the thrust in the rod connecting the locomotive to the carriage.

A child connects two toy wagons together using a light horizontal rod and pushes them along a horizontal level path.

Each wagon has a mass of , and the resistance to motion of each wagon is modelled as a constant force of .

The child pushes the rear-most wagon with a constant horizontal force of and the wagons accelerate at

Find the value of .

Find the thrust in the connecting rod.

Figure 1 shows particles and which have masses of 5 kg and 9 kg respectively.  The particles are connected by a light inextensible string that passes over a smooth light fixed pulley.

The particles are released from rest with the string taut.

Calculate

(i) the initial acceleration of the particles,

(ii) the tension in the string as descends.

Figure 1 shows two particles and which have masses of 3 kg and kg respectively. The particles are connected by a light inextensible string that passes over a smooth light fixed pulley.

The particles are released from rest with the string taut and particle begins to accelerate downwards at a rate of .

Find the value of .

A tugboat is pulling a barge across the horizontal surface of the water in a harbour. A horizontal towrope is used to connect the tugboat to the barge.

The barge has a total mass of 2 900 tonnes, and as it moves through the water it experiences a constant resistive force of 5 000 N.

The barge is initially moving at a speed of . After accelerating at a constant rate for 3 minutes, its speed reaches   The direction of the barge’s acceleration is the same as the direction of its velocity at all times.

For the period of time when the barge is accelerating, find the tension in the towrope.

A particle is acted upon by two forces,  and ,  given in vector form as

Given that the particle has a mass of 800 g, find the magnitude of the acceleration experienced by the particle under the combined action of the two forces.

A particle is acted upon by two forces, and ,  given in vector form as

Given that the particle has a mass of 1160 g, find the magnitude of the acceleration experienced by the particle under the combined action of the two forces.

A ball of mass is attached to one end of a string.

The other end of the string is attached to a ball of mass .

The string passes over a fixed pulley.

The system is held at rest with the balls hanging freely and the string taut.

The hanging parts of the string are vertical with at a height above horizontal ground and with at a height above the ground, as shown in Figure 1.

The system is released from rest.

In the subsequent motion, does not rebound when it hits the ground and does not hit the pulley.

The balls are modelled as particles.

The string is modelled as being light and inextensible.

The pulley is modelled as being small and smooth.

Air resistance is modelled as being negligible.

Using this model,

(i) write down an equation of motion for ,

(ii) write down an equation of motion for .

Using this model, find, in terms of only, the height above the ground at which first comes to instantaneous rest.

State one limitation of modelling the balls as particles that could affect your answer to part (b).

In reality, the string will not be inextensible.

State how this would affect the accelerations of the particles.

Figure 1 shows a box of mass 2 kg resting on a rough horizontal table. It is connected by a light inextensible string to a metal sphere of mass 1 kg. The string passes over a smooth light fixed pulley at the edge of the table so that is hanging vertically downwards.

The string between and the pulley is horizontal, and the magnitude of the frictional force between and the table is 7.7 N.

The system is released from rest with the string taut.

Calculate

(i) the initial acceleration of the two objects,

(ii) the tension in the string as descends.

After descending for 1.5 seconds, sphere strikes the ground and immediately comes to rest. 

When strikes the ground, box  is exactly 14 cm from the pulley.

Determine if  will strike the pulley before friction causes it to come to rest.

Show your full working.

Figure 1 shows two masses,  and , in a light scale-pan. Mass  rests on top of mass .

Mass  has a mass of  700 grams  and mass  has a mass of  900 grams. 

The scale-pan is attached to a vertical light inextensible string.

Using the string, the scale-pan is raised vertically with an acceleration of

Find the tension in the string.

Find the magnitude of the force exerted on mass  by mass .

Find the magnitude of the force exerted on mass  by mass .

Find the force exerted on mass  by the scale-pan.

A cheese company stores its cheeses in an underground cave. The cheeses are taken in and out of the cave using a vertical lift.

The lift consists of a horizontal pallet attached to the lift mechanism by two identical support ropes. The tensions in the two ropes are kept equal to each other at all times.

The combined mass of the pallet and a full load of cheeses is 1700 kg.

While being lowered, the pallet initially experiences a constant vertical acceleration downwards. The pallet starts at rest, and after 4 seconds it has moved a total of 20 metres.

Find the tension in each of the two ropes, during this initial motion.

The ropes attached to the pallet can each safely withstand a force of up to 10 400 N without breaking.

A new motor is installed in the lift mechanism. At maximum power it would be able to raise a 1700 kg load 30 metres in 4.9 seconds, with the load starting at from rest and experiencing a constant acceleration throughout.

Determine if the new motor can safely be used at maximum power to raise a full pallet of cheeses out of the cave. 

Show full working.

A weather balloon is connected to a scientific instrument by a light inextensible cable. The balloon has a mass of 600 grams and the instrument has a mass of 2 kg. 

The balloon is released from rest with the cable taut and it rises into the sky. During the initial period of ascent the balloon is at all times directly above the instrument, with the balloon and instrument experiencing a constant upwards acceleration.

Other than gravity and the upward lift provided by the weather balloon, all other external forces on the balloon and instrument may be ignored.

Given that the tension in the cable is during the initial period of ascent, find the upward lift provided by the weather balloon during the initial period of ascent.

Figure 1 shows a lift that is used to raise a large crate.

The crate has a mass of 1200 kg and the platform of the lift has a mass of 300 kg. The upward force is transmitted to the platform through a light vertical rod.

Starting from rest, the platform and crate are accelerated vertically upwards at a constant rate. They reach a velocity of after 2 seconds.

Find the thrust in the rod.

Find the force exerted by the crate on the lifting platform.

Figure 1 shows a small block of mass 4.5 kg resting on a rough horizontal table. Block is attached to a small metal sphere of mass 3.5 kg by a light, inextensible string.

The string passes over a small, smooth, fixed pulley at the edge of the table so that the portion of the string between and the pulley is horizontal and the portion between the pulley and is vertical.

The resistance to motion between block and the table is modelled as a constant horizontal force of magnitude .

Initially the string is taut and sphere hangs 1.0 m above the ground. The system is released from rest, and 0.80 s later sphere strikes the ground.

Determine the value of .

A particle of mass 3 kg starts from rest and is acted upon by three forces, , and ,  given in vector form 

where  is a constant.  The resultant of forces  ,  and  is .

Given that  acts on a bearing of , find the value of the constant .

Work out the magnitude of the acceleration of the particle.

Find the total distance travelled by the particle in the first 4 seconds of its motion.

A particle of mass 5 kg starts from rest and is acted upon by three forces, ,  and , given in vector form as

where  is a constant.  The resultant of forces ,  and is .

Given that acts on a bearing of 135^° find the value of the constant .

Find the total distance travelled by the particle in the first 3 seconds of its motion.

After the first 3 seconds a new force is applied to the particle, in addition to forces ,  and .  After the addition of the new force the particle then moves with a constant velocity in the direction of . 

Write down the value of in vector form.

A particle is acted upon by two forces,   and ,  given in vector form as

     and      

where and are perpendicular unit vectors.

Given that the particle experiences an acceleration of magnitude   under the combined action of the two forces, find the mass of the particle.

In the Extreme Well Challenge, a bucket of water with a mass of kg must be raised out of a well that is m deep. The bucket is raised by means of a light rope that remains vertical at all times. The bucket begins at rest at the bottom of the well, and for the first m of the ascent it must accelerate at a constant rate of   After that the rope must be allowed to go slack so that gravity is the only force operating on the bucket.  The distance must be chosen so that the velocity of the bucket becomes momentarily zero just as it reaches the top of the well.

The time taken to complete the Extreme Well Challenge is measured from the moment the bucket begins accelerating upwards from the bottom of the well, until the moment that the bucket’s velocity becomes momentarily zero at the top of the well.

Show that the time required to complete the Extreme Well Challenge for a well metres deep is given by

where is measured in seconds, is the constant acceleration of the bucket during the first part of its ascent, and is the acceleration due to gravity.

You have bought a new rope and need to know the maximum safe tension that the rope can withstand without breaking. 

The manufacturer tells you that:

“operating at its maximum safe tension this rope could be used to complete the Extreme Well Challenge, with a 30 metre deep well and a 10 kilogram bucket of water, in 4.20 seconds”

Hence determine the maximum safe tension that your new rope can withstand without breaking, giving your answer in newtons correct to 3 significant figures.

In the mystical kingdom of Newtonia, a unicorn is using its horn to push a large stone of power across the icy ground towards the spot known as the Point of Destiny. 

The mass of the stone is 4600 kg, and the ground is perfectly level.  As the stone moves across the ground it experiences a constant resistance to movement of 3200 N. 

The unicorn’s horn, which may be modelled as a light rod, is held horizontal at all times.  While the unicorn’s magic would allow it to push with almost any force, the maximum thrust which the unicorn’s horn can withstand without shattering is 100 kN.

When the stone is exactly 100 metres away from the Point of Destiny and is being pushed at a speed of 1.8 metres per second, an evil wizard appears and begins to cast a spell of doom.  The spell will take exactly 3 seconds for the wizard to cast.

Given that the acceleration of the stone must remain constant over the entire 100 metre distance, and that the unicorn’s horn must not be allowed to shatter, determine whether or not the unicorn can get the stone to the Point of Destiny before the wizard completes his spell. 

Show complete mathematical workings to support your answer.

A large rock with a mass of 230 kg is tied to a buoy by a light inextensible rope and is then thrown into the water.  The rock is heavy enough so that as it sinks vertically downwards it pulls the buoy through the water behind it. 

As the rock sinks through the water, it experiences a resistance to its motion that may be modelled as a constant upwards force of 150 N. 

The buoy has a mass of 10 kg, and the combination of its buoyancy and the water resistance to its movement may be modelled by a constant upwards force of N.

Given that the tension in the rope as the rock sinks is 655 N, determine the value of .

A train locomotive of mass 10 000 kg is used to pull carriages along a section of horizontal track. 

The connection between the locomotive and the first carriage, and the connections between each carriage, may all be modelled as light rods parallel to the direction of motion along the track. 

The locomotive’s engine is able to provide a maximum driving force of 20 000 N, and the resistance to motion of the locomotive is modelled as a constant force of 3000 N. 

The carriages all have the same mass, and the resistance to motion of each carriage is modelled as a constant force of 1000 N.

A guard in the rearmost carriage of the train has been measuring the tension in the connecting rod between the rearmost carriage and the one in front of it.  He has noticed that with the train moving forward and the locomotive providing maximum driving force in the forward direction, the tension measured when there are three carriages attached to the locomotive is 437.8 N greater than the tension measured when there are four carriages attached to the locomotive.

Use this information to find the mass of a single train carriage, giving your answer correct to 4 significant figures.

A lift which has a mass of 250 kg when empty is being raised vertically using a light inextensible cable attached to its top.

Inside the lift is a horizontal platform with mass 3 kg, which is connected to the floor of the lift by a light  vertical rod.

On top of platform is a light wire frame from which hangs a metal sphere of mass 50 grams, supported by a light inextensible string attached to the top of the frame.  This scenario is shown in Figure 1 below.

Starting from rest, the lift is accelerated upwards with a constant acceleration of magnitude .

Given that the thrust in the rod connecting to the floor of the lift is 35.38 N, find the tension in the string connecting to the wire frame.

Find the tension in the lift cable.

Find the distance that the lift ascends in its first 3 seconds of motion.

Figure 1 shows two particles and with masses of  kg and 12 kg respectively.  The particles are connected by a light inextensible string that passes over a smooth light fixed pulley.

The particles are released from rest with the string taut and one of the particles begins to accelerate downwards at a rate of  .

Find the possible values of .

Given that the tension in the string is 134.4 N, find the precise value of . Show your full reasoning.

Figure 1 shows a block of mass  kg resting on a rough horizontal table. It is connected by a light inextensible string to a metal sphere of mass  kg.  The string passes over a smooth light fixed pulley at the edge of the table so that is hanging vertically downwards.

The string between and the pulley is horizontal, and the frictional force between  and the table is modelled as a constant force of magnitude  N.

The system is released from rest with the string taut.

Given that sphere begins to descend after the system is released, show that as descends the tension in the string, , is given by

where is the acceleration due to gravity.

Given that sphere remains motionless when the system is released, find an expression for   in terms of and the acceleration due to gravity .

A particle of mass 110 grams starts from rest and is acted upon by three forces, ,   and ,  given in vector form as

where and are constants with  . 

Initially the particle is at rest and located at the origin.

Given that the position vector of the particle is    at time seconds,  where  ,  find the values of  , and .