3.1 Newton's Laws of Motion

Three boxes A, B, and C are connected using an inextensible rope of negligible mass, as shown in Fig. 1.1.

Fig. 1.1

Boxes A and B each have a mass of 2.4 kg, and the magnitude of the frictional force  between each box and the surface is equal to 

where  is the normal reaction force and  refers to the box in question A, B or C.

Box C descends with constant velocity.

Calculate the tension in the rope connecting boxes A and B.

Determine the tension in the rope connecting boxes B and C.

Hence, determine the mass of box C.

The rope connecting A and B breaks abruptly.

A rock climber of mass 80 kg slips after attaching a bolt firmly onto a piece of rock that they are climbing as shown in Fig. 1.1. 

At the lowest point of the climber's fall, the length of the rope between the harness and the bolt is 24 m. The climber comes to rest and all the energy from the fall has been absorbed by the rope.

The rope has an unstretched length of 19 m and a stiffness of 560 N m^–1. 

Fig. 1.1

Assume that the extension of the rope is proportional to its tension and the mass of the rope and the effects of air resistance are negligible.

Calculate

Hemp is one of the strongest fibres with a very high tensile strength and is extremely rigid. Hemp ropes were the first to be used for ship rigging which adjusted the position of the sails and supported the masts. 

However, rock climbers use nylon ropes to break their fall as they can stretch considerably under tension.

A climbing equipment company tests their ropes by attaching an 80 kg mass to an end of the rope and releasing it from a ledge. 

Fig. 1.2 shows the results of a test carried out to compare the time it takes for a nylon rope and a hemp rope to return to their unstretched length after beginning to stretch.

Fig. 1.2

In the test, both ropes used had identical unstretched lengths of 19.0 m and diameters of 10 mm.

Show that the average resultant force exerted by the hemp rope is about 20 times larger than the average resultant force exerted by the nylon rope.

Discuss in terms of force and momentum, why nylon ropes are favoured by rock climbers compared to hemp ropes.

A man with a mass of 65 kg stands in an elevator. The elevator initially accelerates uniformly for 1.8 s followed by a period of a constant velocity of 2.5 m s^–1 and then a final uniform deceleration for 1.8 s.

Calculate the magnitude of the reaction force on the man from the elevator floor during the time it takes the elevator to come to rest.

The man uses the elevator again to travel to a lower floor. The elevator descends at the same speed and comes to rest in the same time as in (a).    

Calculate the reaction force on the man from the elevator floor during the time it takes to come to rest on the floor below.

Suggest and explain during the deceleration period, which direction of elevator travel would make the man feel

In a stroke of bad luck, the elevator cable snaps and it falls freely under gravity. During the fall, the man experiences a sense of weightlessness. 

Explain, using appropriate laws of motion, why the man feels a sense of weightlessness in this situation.

Newton’s third law refers to pairs of forces.

For a pair of forces that obey Newton's third law, describe:

Two blocks A and B are joined by a string and rest on a frictionless horizontal table as shown in Fig. 1.1. A force of 200 N is applied horizontally on block B.

Fig. 1.1

Block A has a mass of 3.0 kg and block B has a mass of 7.0 kg. 

The string is removed from the blocks. Block A slides along the horizontal frictionless surface towards block B, which is stationary, as illustrated in Fig. 1.2.

There are no resistive forces acting on block A as it moves towards block B. At time t = 0, block A has a velocity of 0.20 m s⁻¹. A short time later, the blocks collide and then separate.

The variation with time t of the momentum of block B is shown in Fig. 1.3.

Use Fig. 1.3 to describe, without calculation, the magnitude of the acceleration of block B from:

Use Fig. 1.3 to determine

A student uses a motion sensor connected to a laptop to investigate the motion of a hollow ball of mass 1.5 × 10^–2 kg falling through the air.

The ball is dropped from rest. It reaches terminal velocity before it reaches the ground.

The upthrust on the ball is negligible.

Explain briefly why the acceleration of the ball

The variation with time t of the velocity v of the ball as it falls towards the ground is shown in Fig. 1.1.

[2]

Use your answers in (b) to determine the drag on the ball at time t = 0.25 s.

The student adds a small amount of sand inside the hollow ball and repeats the experiment. The ball is dropped from rest and reaches terminal velocity before it reaches the ground.

A cyclist carries out tests on the braking system of their bicycle.

At time t = 0 the cyclist is travelling at an initial speed of u as they apply the brakes and come to a stop at time t = 2.0 s.

Fig. 1.1 shows the variation of the braking force F on the bicycle with time t.

Use Newton’s second law of motion to explain the physical quantity represented by the area under the graph shown in Fig. 1.1.

The total mass of the cyclist and bicycle is 82 kg. 
Use Fig. 1.1 to calculate the initial speed u.

Complete Fig. 1.2 to show the variation of the speed of the bicycle from t = 0 to t = 2.0 s.

Fig. 1.2

Brakes testing is a crucial part of the manufacturing process of any vehicle.

A train manufacturer wants to determine the effect of a rapid deceleration on passengers riding their new model of train.

Four crash test dummies are placed on different seats on the train as shown in Fig. 1.3. During the test, the train travels at a high speed in the direction shown and after a time, the brakes are applied. Seat belts are not used on trains.

With reference to at least one of Newton’s laws of motion, discuss which seat, A−D, is the safest for a passenger to be sitting on in the event of a rapid deceleration.

You may assume that the seats all remain fixed firmly to the floor and do not break.