Question

A rollercoaster cart starts from rest at the top of a 100m high hill. It moves around a frictionless track before entering a vertical, circular loop at ground level. What is the maximum radius that the loop can have if the cart is to clear the top of the loop without slipping? Give your answer to one significant figure, with a unit.​

Answer 1

Answer:

Newtonian mechanics : dynamics of a point mass (1001-1108) - Dynamics of a system of point masses (1109-1144) - Dynamics of rigid bodies (1145-1223) - Dynamics of deformable bodies (1224-1272) - Analytical mechanics : Lagrange's equation...

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Answer 2

To solve this problem, we need to consider the conservation of energy and the centripetal force acting on the rollercoaster cart as it moves around the loop.

At the top of the hill, the cart has potential energy (PE) due to its height above the ground. At the bottom of the loop, this potential energy is converted to kinetic energy (KE) due to the cart's motion. At the top of the loop, the KE is again converted to PE. If the cart is to clear the top of the loop without slipping, it must have enough KE to maintain contact with the track throughout the loop.

Let's assume that the radius of the loop is r. At the top of the loop, the cart is moving horizontally with a speed v, which we can calculate using the conservation of energy:

PE at top of hill = KE at bottom of loop

mgh = (1/2)mv^2

where m is the mass of the cart, g is the acceleration due to gravity (9.81 m/s^2), and h is the height of the hill (100 m).

Solving for v, we get:

v = sqrt(2gh)

At the top of the loop, the centripetal force (Fc) acting on the cart must be equal to the force due to gravity (Fg) in order to maintain contact with the track:

Fc = Fg

mv^2/r = mg

where r is the radius of the loop.

Solving for r, we get:

r = v^2/g

Substituting the value of v we found earlier, we get:

r = (2gh)/g

r = 2h

Substituting the given value of h, we get:

r = 2(100 m) = 200 m

So the maximum radius that the loop can have if the cart is to clear the top of the loop without slipping is 200 m (to one significant figure), with units of meters (m).


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