GOAL Combine gravitational, translational, and rotational energy. PROBLEMA ball
ID: 1494856 • Letter: G
Question
GOAL Combine gravitational, translational, and rotational energy. PROBLEMA ball of mass M and radius R starts from rest at a height of 2.00 m and rolls down a 30.0° slope, as in the figure. What is the linear speed of the ball when it leaves the incline? Assume that the ball rolls without slipping. 2.00 m 30.07 A ball starts from rest at the top of an incline and rolls to the bottom without slipping. STRATEGY The two points of interest are the top and bottom of the incline, with the bottom acting as the zero point of gravitational potential energy. As the ball rolls down the ramp, gravitational potential energy is converted into both translational and rotational kinetic energy without dissipation, so conservation of mechanical energy can be applied.Explanation / Answer
Law of conservation of energy
Total energy at top= Total energy at bottom
consider bottom as the datum, so
total energy at top= potential enery= Mgh
total energy at bottom= linear kinetic energy+ rotational kinetic energy= 1/2Mv2 + 1/2Iw2 =1/2Mv2 +1/2(1/2MR2)(v2/R2) = 1/2Mv2 + 1/4Mv2 = 3/4Mv2
therefore Mgh= 3/4Mv2
gh = 3/4v2
v =( 4gh/3)1/2
v = (4x9.81ms-2x2m/3)1/2
v = 5.1ms-1
linear speedof the ball when it leaves the incline is5.1ms-1
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