Train and one carriage

1
Train and one carriage
2
Connected particles
Horizontal motion of connected particles
Rn1
Rn2
T T
F2
F1
m1g
m2g
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A car and a trailer
A car pulls a trailer along a level road at a
constant velocity v.
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.
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Example
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
5
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
6
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
F2
F1
10g
5g
F1
?R1
39.2
F2
?R2
19.6
? a 4.08ms-2
System as a whole
120 - 39.2 19.6 15a
5 kg particle
T 19.6 5a
? T 40 N
Both particles accelerates at 4.08 ms-2 with a
tension 40 N.
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Rn1
Rn2
T T
10 kg
5 kg
120 N
F2
F1
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.
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Train and two carriages
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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.
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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