Dynamics Exam Questions Q1.

A small ball, P, of mass 0.8 kg, is held at rest on a smooth horizontal table and is attached to one end of a thin rope.

The rope passes over a pulley that is fixed at the edge of the table.

The other end of the rope is attached to another small ball, Q, of mass 0.6 kg, that hangs freely below the pulley.

Ball P is released from rest, with the rope taut, with P at a distance of 1.5 m from the pulley and with Q at a height of 0.4 m above the horizontal floor, as shown in Figure 1.

Ball Q descends, hits the floor and does not rebound.

The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth.

Using this model,

(a) show that the acceleration of Q, as it falls, is 4.2 m s–2

(5)

(b) find the time taken by P to hit the pulley from the instant when P is released.

(6)

(c) State one limitation of the model that will affect the accuracy of your answer to part (a).

(1)

(Total for question = 12 marks)

Q2.

A lift of mass 200 kg is being lowered into a mineshaft by a vertical cable attached to the top of the lift. A crate of mass 55 kg is on the floor inside the lift, as shown in Figure 2. The lift descends vertically with constant acceleration. There is a constant upwards resistance of magnitude 150 N on the lift. The crate experiences a constant normal reaction of magnitude 473 N from the floor of the lift.

(a) Find the acceleration of the lift.

(3)

(b) Find the magnitude of the force exerted on the lift by the cable.

(4)

(Total for question = 7 marks)

Q3.

A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N. The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and R newtons respectively.

Given that the acceleration of the car and the caravan is 0.88 m s−2,

(a) show that R = 860,

(3)

(b) find the tension in the tow-bar.

(3)

(Total 6 marks)

Q4.

A woman travels in a lift. The mass of the woman is 50 kg and the mass of the lift is 950 kg. The lift is being raised vertically by a vertical cable which is attached to the top of the lift. The lift is moving upwards and has constant deceleration of 2 m s−2. By modelling the cable as being light and inextensible, find

(a) the tension in the cable,

(3)

(b) the magnitude of the force exerted on the woman by the floor of the lift.

(3)

(Total 6 marks)

Q5.

A vertical rope AB has its end B attached to the top of a scale pan. The scale pan has mass 0.5 kg and carries a brick of mass 1.5 kg, as shown in Figure 1. The scale pan is raised vertically upwards with constant acceleration 0.5 m s–2 using the rope AB. The rope is modelled as a light inextensible string.

(a) Find the tension in the rope AB.

(3)

(b) Find the magnitude of the force exerted on the scale pan by the brick.

(3)

(Total for question = 6 marks)

Q6.

Figure 5

Three particles A, B and C have masses 3m, 2m and 2m respectively. Particle C is attached to particle B. Particles A and B are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 5. The system is released from rest and A moves upwards.

(a) (i) Show that the acceleration of A is g⁄7

(ii) Find the tension in the string as A ascends. (7)

At the instant when A is 0.7 m above its original position, C separates from B and falls away. In the subsequent motion, A does not reach the pulley.

(b) Find the speed of A at the instant when it is 0.7 m above its original position.

(2)

(c) Find the acceleration of A at the instant after C separates from B.

(4)

(d) Find the greatest height reached by A above its original position.

(3)

(Total 16 marks)

Q7. A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find

(a) the acceleration of the car and trailer,

(3)

(b) the magnitude of the tension in the towbar.

(3)

The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude F newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

(c) find the value of F.

(7)

(Total 13 marks)

Examiner's Report Q1. Although the first part of this question was well done by some candidates, a significant minority failed to write down an equation of motion of the correct structure for each mass. Some included a weight term in the equation for the particle moving horizontally whilst others showed no real understanding of what was required and confused equations of motion with suvat equations or omitted this part completely. Those who managed to produce two valid equations generally had no difficulty in deriving the given answer for the acceleration. Since it was a 'show that' question, it was important that the final answer followed from sufficient and entirely correct working; those who went straight from the two simultaneous equations to writing down the answer were not awarded the final mark. Such instances were, however, fairly rare. Some attempted a 'whole system' approach which was equivalent to writing down just one equation with the tension eliminated. This was deemed not sufficient for a complete method and scored a maximum of two marks. Part (b) was also completed with mixed degrees of success. To solve the problem it was necessary to split the motion into two stages, before and after one mass hits the ground. Those who attempted to use a single suvat equation to cover the whole motion could achieve no credit. Some successfully found the time it took for Q to fall to the ground but were then unsure how to proceed. Those who calculated the speed with which Q hit the ground generally used this in considering the subsequent motion of P as it continued to move across the table. However, the most common error at this stage was in not realising that P would move with constant speed since there was no tension and the table surface was smooth; some included an acceleration of 4.2 ms-2 (or even 9.8 ms-2) in the relevant suvat equation or, more commonly, used s = ½ (u + v)t with v = 0. These candidates generally scored three out of a possible six marks. The final part required the identification of a possible limitation of the model. The assumptions of this model were listed in the question and so any one of these (such as smooth pulley, light string, inextensible string, modelling the balls as particles) could be used in describing such a limitation. Often other factors were identified which were not part of the specified assumptions for this model and so were not awarded the mark. Q2.

This question proved to be a real discriminator with many failing to appreciate how internal and external forces work, it was not unusual for students to produce incorrect equations and score zero marks for the whole question. Future students would be advised to spend time analysing lift systems and practising writing down the equations of motion for each part of the system. In part (a), the motion of the lift was attributed with two unknown quantities and hence the first part of the question could only be done in one stage by considering the forces acting on the crate. Students able to arrive at the correct value for the acceleration were very much in the minority. The second part could be done without using the answer to part (a), and most successful attempts came from using a whole system equation. Those who used an equation for the lift only, tended to be unsuccessful. Having arrived at the correct answer, a good proportion then forgot that this question involves the use of g and neglected to write the final answer to either two or three significant figures thus losing the final mark. Other errors usually involved the omission of m from one or both terms, mixing the two masses and including an extra g in all of the terms.

Q3.

In the first part, the majority of candidates wrote down a correct equation of motion for the 'whole system' which they successfully solved to derive the given value for the resistance force on the caravan. Some chose to consider the car and caravan separately, calculating the tension from the car equation and then using this value in the caravan equation, again generally successfully. There were more errors evident in finding the tension in part (b); the mass used in the 'ma' term was not always consistent with the rest of the equation and occasionally

the mass of the whole system was used in an equation relating only to one body. Sometimes the two resistances were confused, two tensions were added together in one equation or the 'ma' term was omitted completely, showing a lack of understanding of the motion of connected particles. Less significant errors tended to involve wrong signs. Overall, however, this question was very well done with full marks often awarded.

Q4.

Part (a) was mostly well done with the vast majority attempting an equation of motion for the whole system. The most common error was the omission of the minus sign on the acceleration. The second part proved to be a discriminator and revealed a lack of understanding of the basic principles. A significant number treated it as a statics problem, even though they had used an acceleration in part (a), and tried assuming the forces were in equilibrium. Amongst those who did attempt to write down an equation of motion for the woman alone, there was much confusion over which forces were acting on her, with many including the tension in the lift cable.

Q5.

This question, particularly part (b), proved to be a real discriminator with many failing to appreciate how internal and external forces work. In part (a) the majority of students were able to use the whole system to calculate the value of the tension. However there were still some students who confused mass and weight on both sides of the equation. The second part was much more problematic with many students not knowing which forces to include in their equation and of those that did, a significant number lost a mark by giving the final answer to 4 SF.

Q6.

This question was well answered by the majority of students. In question (a) most identified correctly individual equations of motion for the two masses and then solved them simultaneously to find the acceleration. Since the answer was given, any potential sign errors tended to be rectified but occasionally the answer did not strictly follow from the working. Sometimes the values '3' and '4' were used rather than ' 3m ' and ' 4m ' as given in the question. This was penalised as accuracy errors here, but all subsequent marks for the rest of the question were available. The most common error in finding the tension was to omit ' m ' in the final answer despite it being included in the working.

In the second part nearly all students found the velocity correctly by using 'v2 = u2 + 2as'; the only significant error seen was in using ' g ' rather than 'g⁄7' showing a lack of understanding of the situation

Question (c) required a similar approach to question (a) but with one different mass. Since the answer was not given this time, there were some arithmetic and sign errors, but generally it was well done. Those who used a value of the tension from question (a) achieved no credit, as did those who tried to somehow use constant acceleration formulae.

The majority of students used an appropriate constant acceleration formula in the final part to find the maximum height reached, using the values of velocity and acceleration from previous parts of the question. Occasionally 'g⁄7' or ' g ' were used. Most, but not all, added '0.7' from the initial part of the motion to reach the final answer as required.

Full marks for this question were often achieved and much good working was seen.

Q7. Part (a) was well done by the majority of candidates and a good number went on to use the answer correctly in part (b). If mistakes were made they were the usual sign errors or more seriously, in terms of marks lost, missing terms.

The third part was poorly done. There was confusion over the direction of the forces and the concept of thrust. A few candidates halved the thrust and used 50N in each equation. Some used the values of the acceleration and tension from previous parts.

Mark Scheme Q1.

Q2.

Q3.

Q4.

Q5.

Q6.

Q7.