Newton's Second Law (CIE A Level Maths: Mechanics)

In each of the scenarios depicted below, the forces acting on the body cause it to accelerate as shown.  The acceleration due to gravity is indicated by .

Find the value of the unknown variable  –  acceleration (), mass () or force ()  –  in each case.

As part of a stage show, a singer is lifted up into the air by a cable attached to a special harness.  The singer has a mass of 54 kg, and at the start of the lift is standing stationary on the floor.  The singer accelerates vertically upwards at a constant rate, and after 2 seconds has reached a height of 3 metres.

By using the appropriate  formula, determine the acceleration experienced by the singer during the lift.

Find the tension in the cable during the lift.

After singing the show’s grand finale, the singer 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 while the singer was being lifted.

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

A child is pulling a cart along a horizontal path by means of a horizontal rope attached to its front end.  The cart has a total mass of 15 kg, and as it moves it experiences a constant resistance to motion of magnitude 2 N.  The cart starts at rest and accelerates at a constant rate, and after 5 seconds it has reached a speed of 2 m s^-1.

Find:

the acceleration of the cart

the tension in the rope.

Two particles A and B are connected by a light inextensible string.  Particle  has a mass of 5 kg, particle  has a mass of 15 kg, and particle hangs directly below particle A.  A force of  300 N  is applied vertically upwards on particle ,  causing the particles to accelerate.

Find:

the magnitude of the acceleration

the tension in the string.

A train locomotive of mass 7000 kg and a carriage of mass 2000 kg are at rest on a section of horizontal track.  The connection between the locomotive and carriage may be modelled as a light rod parallel to the direction of their motion forward or backward along the track.  The resistances to motion of the locomotive and the carriage are modelled as constant forces of 2300 N and 1000 N respectively.

The locomotive begins to accelerate in the backwards direction, with its engine providing a constant driving force of 15000 N.

Find:

the magnitude of the acceleration of the locomotive and carriage

the thrust in the connecting rod.

Two masses  and  are placed in a light scale-pan, with mass  resting on top of mass  as shown in the diagram below:

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

the force exerted on mass  by mass 

the force exerted on mass  by mass 

the force exerted on mass  by the scale-pan.

Two particles A and B have masses of 6 kg and 9 kg respectively.  The particles are connected by a light inextensible string that passes over a smooth light fixed pulley as shown in the diagram below:

The particles are released from rest with the string taut.

Calculate the acceleration of the particles and the tension in the string as B descends.

A box B of mass 2 kg rests on a rough horizontal table. It is connected by a light inextensible string to a metal sphere A of mass 1 kg. The string passes over a smooth light fixed pulley at the edge of the table so that A is hanging vertically downwards as shown in the diagram below:

The string between B and the pulley is horizontal, and the magnitude of the frictional force between B and the table is 7.6 N.

The system is released from rest with the string taut.

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

After descending for 1.5 seconds, sphere A strikes the ground and immediately comes to rest.  At that moment, box  is exactly 14 cm from the pulley.

By first calculating the speed of  at the moment  hits the ground, and then employing an appropriate equation of motion, determine whether or not  will strike the pulley before friction causes it to come to rest.

In each of the scenarios depicted below, the forces acting on the body cause it to accelerate as shown.  The acceleration due to gravity is indicated by g. 

Find the value of the unknown variable  –  acceleration (a), mass (m) or force (P)  –  in each case.

To improve the flavour of its products, an artisan cheese company stores its cheeses in a deep cave. The cheeses are brought in and out of the cave by means of a vertical lift, consisting of a horizontal pallet attached to the lift mechanism by two support ropes.  The tensions in the two ropes are kept equal to each other at all times.  A full pallet of cheeses has a total mass of 1700 kg

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

Determine the acceleration of the pallet, and the tension in each of the two ropes, during this initial portion of the descent.

The hemp support ropes attached to the pallet can each safely withstand a force of 10.4 kN 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 rest and experiencing a constant acceleration throughout the time.

Determine whether or not the new motor can safely be used at maximum power to raise a full pallet of cheeses out of the cave.  Be sure to show full mathematical workings to support your answer.

A tugboat is pulling a barge across the horizontal surface of the water in a harbour, by means of a horizontal towrope connecting the tugboat to the barge. The barge has a total mass of 2900 tonnes, and as it moves through the water it experiences a constant resistance to movement of 5000 N. The barge is initially moving at a speed of ,  and after accelerating at a constant rate for 3 minutes its speed has increased to   The direction of the barge’s acceleration is at all times the same as the direction of its velocity.

Find the tension in the towrope during the time that the speed of the barge is increasing.

A weather balloon is connected to a bundle of scientific instruments by a light inextensible cable.  The balloon has a mass of 600 g and the bundle of instruments has a mass of 2 kg.  The balloon is released from rest with the cable taut, and during the initial period of ascent the balloon is at all times directly above the bundle, with the balloon and bundle experiencing a constant upwards acceleration.  Other than gravity and the lift provided by the weather balloon, all other external forces on the balloon and bundle may be ignored.

Given that the tension in the cable is during the initial period of ascent, find

during the initial period of ascent.

A child has connected two toy wagons together with a light horizontal rod and is pushing them along a horizontal level track.  Each wagon has a mass of , and the resistance to motion of each wagon is modelled as a constant force of P N.

The child pushes on the rearmost wagon with a constant horizontal force of and the wagons experience a forward acceleration of

Find:

the value of the constant P

the thrust in the connecting rod.

In a warehouse a hydraulic lift is being used to raise a large crate as shown in the diagram below:

The crate has a mass of 1200 kg and the platform of the lift has a mass of 300 kg. The lifting force is transmitted to the platform by means of a vertical rod of negligible mass.

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

Find: the thrust in the rod

the force exerted by the crate on the lifting platform.

Two particles A and B have masses of 3 kg and m_B kg respectively. The particles are connected by a light inextensible string that passes over a smooth light fixed pulley as shown in the diagram below:

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

Find the value of m_B.

A scientist has set up a device to measure the frictional resistance force between a block B and a rough horizontal tabletop.  Block B has a mass of 4.5 kg, and it is connected by a light inextensible string to a metal sphere A of mass 3.5 kg.  The string passes over a smooth light fixed pulley at the edge of the table so that A is hanging vertically downwards as shown in the diagram below:

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

The system is released from rest with the string taut and with sphere A exactly 1 metre above the ground.

Given that sphere A strikes the ground 0.8 seconds after it is released, determine the value of F_f.

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 only 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”.

In the Extreme Well Challenge, a bucket of water with a mass of m kg must be raised out of a well that is h 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 x 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 x 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 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 T required to complete an Extreme Well Challenge for a well h metres deep is

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

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 required force, the maximum thrust which the unicorn’s horn may 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.  Be sure to 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 P N.

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

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 the different carriages, 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 a single 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 that 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 small lift car with an empty mass of 250 kg is being raised vertically by means of a light inextensible cable attached to its top.  Inside the lift car is a horizontal wooden tabletop B with mass 3 kg, which has been connected to the floor of the lift car by a light  vertical rod.  On top of B is a light wire frame from which hangs a metal sphere A of mass 50 g, supported by a light inextensible string attached to the top of the frame.  This situation is depicted in the diagram below:

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

Given that the thrust in the rod connecting B to the floor of the lift car is 35.38 N, find:

the tension in the string connecting A to the wire frame

the tension in the lift cable

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

Two particles A and B have masses of m_A kg and 12 kg respectively.  The particles are connected by a light inextensible string that passes over a smooth light fixed pulley as shown in the diagram below:

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 m_A.

Given that the tension in the string is 144 N, find the precise value of m_A.

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

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

The system is released from rest with the string taut.

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

where g is the acceleration due to gravity.

Given that the sphere A remains motionless after the system is released, find an expression for F_f  in terms of m_A and the acceleration due to gravity g.