Who's The Fastest?

A student is planning to do an incline plane experiment. Using identical incline planes, the student will roll down a metal hoop on one, and a block with frictionless ball bearing rollers on the other. (See diagram below). Both the hoop and block have the same mass. Before doing the experiment the student asks other students which object will have the greater velocity when it reaches the bottom of its ramp. Their opinions are below. Which one is correct?

• A. One student says the final velocities will be the same since both objects have the same initial gravitational potential energy, and with no friction losses, all the potential energy will be converted to kinetic energy and both objects will have the same final kinetic energy and same final velocity.
• B. A second student says that you really can't tell since the final velocity of the hoop depends on the radius of the hoop.
• C. A third student says that the hoop will have the greater final velocity because even though it starts slower, it will speed up at a greater rate and end with a larger final velocity than the block.
• D. The fourth student says that the first student is nearly right, but the block will have a greater final linear velocity since its kinetic energy is all linear, but the hoop's kinetic energy is both linear and rotational, giving the hoop a smaller final (linear) velocity than the block?

See if you picked correctly.

Answer

Who's The Fastest? - Answer = D

Answer: The correct answer is D. Students one and four are correct that both object have the same final kinetic energy. The block's is all linear so we have: mgh = 1/2mv^2, the mass cancels and we find a final velocity for the block of v = square root(2gh). The hoop has both linear and rotational kinetic energy, so we have mgh = 1/2mv^2 + 1/2 Iw^2, (where I = moment of inertia = mr^2, and w = angular velocity = v/r). Expressing the angular quantities in linear terms we have: mgh = 1/2mv^2 + 1/2 (mr^2)(v/r)^2. Canceling terms we get mgh = 1/2mv^2 + 1/2 mv^2 (The energy is divide equally between linear and rotational kinetic energy). Canceling the mass, combining terms, and solving for the final velocity of the hoop we get: v = square root(gh). This number is small than the velocity of the block. [sqrt(2gh) versus sqrt(gh)}]. Thus the block has the greater final linear velocity, and also reaches the bottom first. (Notice that the final velocity of the hoop does not depend on its radius.)

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