Train and one carriage

Connected particles Horizontal motion of connected particles Rn1 Rn2 T T F2 F1 m1g m2g

A car and a trailer A car pulls a trailer along a level road at a constant velocity v. T The trailer is pulled forward by the tension T in the tow bar. The trailer will exert an equal and opposite force T on the car. If the car is accelerating , then there must be a force acting on the trailer to produce this acceleration, and this will be provided by the tension T in the tow bar. On the other hand, when the car is slowing down, so is the trailer. In the absence of brakes on the trailer, some force must act in the opposite direction to the motion of the car and trailer. In this case the tow bar will exert thrust on both the car and the trailer.

Example T R1 R2 m2 g m1 g a ms-2 F The diagram shows a car of mass m1 pulling a trailer of mass m2 along a level road. The engine of the car exerts a forward force F, the tension in the tow bar is T and the reactions at the ground for the car and the trailer are R1 and R2 respectively. If the acceleration of the car is a, write down the equation of motion for: Solution (a) F = (m1 + m2)a (b) F – T = m1a (c) T = m2a (a) the system as a whole, (d) R1 = m1g and R2 = m2 g (b) the car, (c) the trailer (d) R1 and R2

Example Solution Car: 2200 – T – 200 = 1100a Caravan: T – 100 = 800 1 1100 g 1 ms-2 2200 N 200 N 800 g 100 N Example A car of mass 1100 kg tows a caravan of mass 800 kg along a horizontal road. The engine of the car exerts a forward force 2.2 k N. The resistance to the motion of the car and caravan are 200 N and 100 N respectively. Given that the car accelerates at 1 ms-2 find the tension in the tow bar. Solution Car: 2200 – T – 200 = 1100a Caravan: T – 100 = 800 1 2200 – 200 – 1100 = T So T = 900 N So T = 900 N

Example Two particles of mass 5 kg and 10 kg are connected by an inextensible string. The particle of mass 10 kg is being pulled by a horizontal force of 120 N along a rough, horizontal surface. Given that the coefficient between each particle and the surface is 0.4, find the acceleration of the system and the tension in the string. Rn1 Rn2 T T 10 kg 5 kg 120 N F1 F2 10g 5g F1 = R1 = 39.2 F2 = R2 = 19.6 System as a whole: 120 - 39.2 – 19.6 = 15a  a = 4.08ms-2 5 kg particle: T – 19.6 = 5a  T = 40 N Both particles accelerates at 4.08 ms-2 with a tension 40 N.

Rn1 Rn2 10 kg T T 5 kg 120 N F1 F2 10g 5g 10 kg particle: 5 kg particle: R1 – 10g = 0 R2 – 5g = 0 R1 = 98 R2 = 49 F1 = R1 = 0.4  98 = 39.2 F2 = R2 = 0.4  49 = 19.6 120 – 39.2 – T = 10a [1] T – 19.6 = 5a [2] [1] + [2] 120 - 39.2 – 19.6 = 15a  a = 4.08ms-2 5 kg particle: T – 19.6 = 5a  T = 40 N Both particles accelerates at 4.08 ms-2 with a tension 40 N. Both particles accelerates at 4.08 ms-2 with a tension 40 N.

Train and two carriages

Find the acceleration of the train and tensions T1 and T2. A train consists of an engine of mass 60 000 kg coupled to two trucks A and B of masses 10 000 kg and 8 000 kg respectively. The couples are light, rigid and horizontal. The train moves along a horizontal track with a constant acceleration. The resistance to motion of the engine, truck A and truck B are 12 000 N, 6 000 N and 8 000 N respectively. The engine exerts a driving force of 45500 N. T2 T2 T1 T1 8000kg 10000kg 60000kg 45500 N 8000N 6000 N 12000 N Find the acceleration of the train and tensions T1 and T2.

System as a whole: 45500 – 12000 – 6000 – 8000 = 78000a  a = 0.25 ms-2. Train: 45500 – T1 – 12000 = 60000a  T1 = 18 500 N Truck B: T2 – 8000 = 8000a  T2 = 10 000 N