EXAMPLE 8.12 A Ball Rolling Down an Incline GOAL Combine gravitational, translational, and rotational energy. A ball starts from rest at the top of an incline and rolls to the bottom without slipping. PROBLEM A 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. 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. SOLUTION Apply conservation of energy with (KEt + KEr + PEg), = (KE! + KEr + PEg)f PE = PEg, the potential energy associated with gravity. expressions, noting that 2 2 5 (KEI), = (KEr), = 0 and (PEg)f = 0. Substitute the appropriate general 0 + 0 + Mgh = inz + 1 ( 2MR2)m2 + 0 The ball rolls without slipping, so Mgh = inz + 1 M122 in2 Rd) = v, the "no-slip condition," can be 2 5 10 appHed. Solve for v, notin that M cancels. 10 h 2 9 v = '/ g = .\\/10(9.80 m/s )(2.00 m) = 529 m/s 7 7 LEARN MORE QUESTION Rank from fastest to slowest: (1) a solid ball rolling down a ramp without slipping, (2) a cylinder rolling down the same ramp without slipping, (3) a block sliding down a frictionless ramp with the same height and slope. (Select all that apply.) C] The cylinder is fastest. C] The ball is slowest. C] The ball is fastest. [:1 The block is fastest. C] The block is slowest. [:1 The cylinder is slowest. PRACTICE IT Use the worked example above to help you solve this problem. A ball of mass M and radius R starts from rest at a height of h = 1.80 m and rolls down a 6 = 37.0 slope, as shown in the figure. What is the linear speed of the ball when it leaves the incline? Assume that the ball rolls without slipping. |:l m/s EXERCISE HINTS: GETTINGSTARTED | I'M STUCK! Repeat this example for a solid cylinder of the same mass and radius as the ball and released from the same height. valinder = :] m/S In a race between the two objects on the incline, which one would win? 0 The ball would win. 0 The cylinder would win. 0 They would tie