Tuesday, 16 July 2013 12:49

## D1-53 LOOP-THE-LOOP

• ID Code: D1-53
• Purpose: Demonstrates centripetal force and conservation of energy in a rotating object
• Description:
This track can be described as three segments: the long upright segment, the loop, and the shorter upright segment. If you begin by placing the ball on the long upright segment at a height equal to the height of the loop (2R), the ball will roll down the track, begin to climb the loop, and then fall off and roll away. You can then repeat this at increasingly higher positions until the ball makes it all the way around the loop and begins to climb the shorter upright segment. In either case, be ready to catch it as it falls off afterwards!

This is a good demonstration to encourage students to make predictions about its behaviour. Invite students to make arguments about what starting height will allow the ball to complete the full loop. A meter stick can be additionally provided upon request to aid in measuring the height.

##### Background
Motion of the ball down the track and around the loop-the-loop can be described in terms of gravitational potential energy, rotational and translational kinetic energy, and centripetal force. A ball of mass m and radius r must be released at some minimum height h above the bottom point of the track so that it will not leave the track while passing around the loop-the-loop.

In order to stay on the track at the top of the loop the centrifugal reaction of the ball on the track must be equal to or greater than the gravitational force on the ball: mv^2/R = mg, or v^2 = gR, where v is its linear velocity at the top of the loop, R is the radius of the loop, and g is the acceleration of gravity. Conservation of energy dictates that at the top of the loop Iw^2/2 + mv^2/2 +2mgR = mgh, where the moment of inertia of the ball I = 2mr^2/5 and w is the angular velocity of the ball at the top of the loop.

From these considerations we obtain the minimum starting height for the ball above the bottom of the loop-the-loop in order that the ball remain in contact with the track at all times: h = 2.7 R. In the case of an object sliding along a frictionless loop-the-loop, the height would be h = 2.5 R. Marks have been made at the points 2.5 R and 2.7 R. The ball remains in contact with the track at the top of the loop only when the height 2.7 R is reached, demonstrating the effect of the rotation of the rolling ball.

• Availability: Available
• Loc codes: FS2
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