A. How much mechanical energy does the disc have at the top of the ramp? B. What

Question

A. How much mechanical energy does the disc have at the top of the ramp?
B. What is the condition for rolling without slipping that related the linear velocity of the center of the disc to its angular velocity.
C. The total kinetic energy of a rolling object is the linear kinetic energy( about the center of mass) plus the rotational kinetic energy about the center of mass. Given this, write out an expression for conservation of energy.
D. How fast is the disc moving at the bottom of the ramp?
E. Is this velocity greater or less than the speed that would be achieved by a block of the same mass sliding without friction down the ramp? What if the solid disc was replaced with a hollow cylinder (same mass and radius)? 2. A ramp has a maximum height of 3.3m. A solid disc, with a mass of 3.5kg and a radius of 0.3m is placed at the top of the ramp and released from rest. The block rolls down the ramp without slipping. want to know the linear speed of the disc when it is at the bottom of the ramp. We a) How much mechanical energy does the disc have at the top of the ramp? b) What is the condition for rolling without slipping that relates the linear velocity of the center of the disc to its angular velocity. c) The total kinetic energy of a rolling object is the linear kinetic energy (about the center of mass) plus the rotational kinetic energy about the center of mass. Given this, write out an expression for conservation of energy. d) How fast is the disc moving at the bottom of the ramp? Is this velocity greater or less than the speed that would be achieved by a block of the same mass sliding without friction down the ramp? What if the solid disc was replaced with a hollow cylinder (same mass and radius)? e)

Explanation / Answer

(A) ME = PE + KE = m g h+ 0

= 3.5 x 9.81 x 3.3

= 113.3 J  

(B) for rolling without slipping.

v= w r

(c) KE = m v^2 / 2 + I w^2 /2 = m v^2 / 2 + (m r^2 / 2)(v/r)^2 / 2

= m v^2 / 2 + m v^2 / 4

= 3 m v^2 / 4

m g h = 3 m v^2 / 4

(d) v= sqrt(4 g h / 3)

v = 6.57 m/s

(e) speed would be greater if there is only.

(bacause all of PE will get converted into translational KE)

then m gh = m v^2 + m v^2

v = sqrt( g h )

speed will be less.

{ as more rotational KE than the disc.}

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