In the diagram above, a small cart of mass, m cart = 0.500 kg, is attached to a
ID: 1789705 • Letter: I
Question
In the diagram above, a small cart of mass, mcart = 0.500 kg, is attached to a launching mechanism that consists of a massless spring of unknown spring constant, k. The spring is retracted a distance of 7.50 cm. When the spring is released, the cart can then successfully make it around the circular loop, which has a radius of 60.0 cm. Assume there is no friction or air resistance in this problem.
a) What is the minimum normal force (in N) on the cart at the top of the loop if it is able to go around without falling off?
b) What minimum speed (in m/s) does the cart need at the top of the loop to make it around the loop?
c) What is the minimum speed (in m/s) the cart can have after it moves away from the spring?
d) Find the minimum value (in N/m) of the spring constant, k.
e) To what height (in m) above the lowest part of the track is the cart able to get up the ramp on the other side of the loop?
cart c3cExplanation / Answer
(A) N = 0
(b) N + m g = m v^2 / r
m g = m v^2 /r
v = sqrt(0.60 x 9.81) = 2.43 m/s
(c) Applying energy conservation,
m g (2 r) + m v^2 /2 = 0 + m v'^2 /2
v' = sqrt[ (9.81 x 4 x 0.60) + 2.43^2 ]
v' = 5.42 m/s
(d) 0.50 x 5.42^2 /2 = k (0.075^2) / 2
k = 2616 N/m
(e) m v'^2 /2 = m g h
h = 5.412^2 / (2 x 9.81) = 1.50 m