A student releases a small cart at the top of an incline with height H above the floor. The cart experiences very little friction. The student is attempting to cause the cart to go around a vertical loop of radius R without the cart losing contact with the track at the top. The student suggests that the heigt H should equal 2R so that the release height and maximum height of th eloop are the same. However, the student finds that it requires noticably higher hieght than 2R for the cart to go around the loop. Explain why H must be noticably greater than 2R to complete the loop. (Hint: In order for the cart to go around the loop it must have a nonzero velocity at the top of the loop.) answer

Explanation:

In the whole process , potential energy of the cart is converted into kinetic energy . At the top of the vertical loop , the whole of potential energy is regained and kinetic energy becomes zero if we release the cart from a height of 2R because difference of height between lowest and highest point of motion  is 2R .  In that case kinetic energy at top = 0 , velocity v = 0

At the top , weight mg is acting which is providing centripetal force . So cart must have some velocity at the top . If it be v

mv²/R = mg

v = √ gR .

For that purpose , the cart must be released from a height greater than 2R .

The extra height beyond 2R will make the velocity at the top non-zero.

initial speed when it is dropped is 150 ft/s

now at the same moment the acceleration due to gravity is perpendicular to its velocity

so as we know that

now we have centripetal acceleration a = 32 ft/s^2 as its perpendicular to velocity

while tangential acceleration at the same moment will be zero

so

now radius of curvature is given as

so the radius of curvature just after it is dropped is 703.125 ft