9. A toy car of mass m is travelling down a frictionless track towards a "loop the loop". Once the car reaches the

bottom of the loop, its speed remains constant throughout the rest of its motion.

a) Draw two free body diagrams, one showing the forces acting on the car at the bottom of the loop, and one

showing the forces acting on the car at the top of the loop. Write down the resultant centripetal

force in both cases.

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b) Given that the car's speed remains constant as it travels round the loop, show that the difference between

the normal reaction force at the top and the bottom of the loop is given by the expression:

Nbottom - Ntop = 2mg

c) The loop has a radius of 25 cm. Calculate the minimum speed the car needs to be travelling at in

order to remain in contact with the track all the way round the loop.

(Hint: Think about what would happen to the reaction force if the car were to lose contact with the

track)

The car is now moving at the minimum speed required to complete the loop, and it has a weight of 2N.

The diagram below shows the car after it has travelled through an angle 8 around the loop, where v

represents the car's velocity, mg its weight, and N is the normal reaction force.

F

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mg

d) Find an expression for the normal reaction force N in terms of the angle and sketch a graph showing the

relationship between N and for the entire loop. Be sure to label any maxima and minima on your

sketch, as well as providing a suitable numerical scale on both axes.

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