3. A solid cylinder is rolling down a ramp as shown in the picture. The cylinder

Question

3. A solid cylinder is rolling down a ramp as shown in the picture. The cylinder is given an initial tangential velocity of v = 2.12m/s, has a radius of r = 0.135m, and a mass of m = 3.16kg. Assume the cylinder rolls without slipping as it travels along the ramp. The image to the right may be unclear. The vertical distance from the top of the ramp to the bottom of the ramp is 1.75 m. The vertical distance from the top of the ramp to the end of the ramp is 1.23 m. a. What is the cylinder’s initial angular velocity? b. What is the cylinder’s MAXIMUM angular velocity?

Explanation / Answer

3. given cylinder rolls wthout slipping on the ramp

intiial tangential velocity vt = 2.12 m/s

r = 0.135 m

hence

initial speed = vi = vt/2 = 1.06 m/s

initial angular speed, wi = vi/r = 7.85185 rad/s

m = 3.16 kg

now, H = 1.75 m

h = 1.23 m

a. cylinders initial angular velocity wi = 7.85185 rad/s

b. cylinders maximum velocity = v

now, from conservation of energy

0.5*mv^2 + 0.5*(0.5*mr^2)v^2/r^2 = 0.5*m*vi^2 + 0.5(0.5*mr^2)*vi^2/r^2 + mgh

hence

v = 4.148 m/s

hecne w max = v/r = 30.73136849 rad/s

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