Question

X B Dk Example 1 A small block of mass m begins from rest at the top of a curved track at a height h and travels around a circular loop of radius R. There is negligible friction between the block and the track between points A and D, but the coefficient of kinetic friction on the horizontal surface between points D and E is u. The distance between points D and E is x. Answer all of the following questions in terms of the given quantities and fundamental constants.

1. Determine the kinetic energy of the block at point C, at the top of the loop.

After the block slides down the loop from point C to D, it enters the rough portion of the track. The speed of the block at point D is the same as point B, and the speed of the block at point E is half the speed of the block at point D.

2. Determine the amount of work done by friction between points D and E.

3. In terms of the coefficient of friction μ and fundamental constants, determine the acceleration of the bock between points D and E.

Answer 1

1] KE at point C = decrease in potential energy = mg*(h- 2R)

2] speed at point D, v= sqrt(2gh)

Work done by friction = change in KE from D to E = 0.5m*(v/2)^2 - 0.5mv^2 = -0.5*m*3v^2/4 = - 0.5*m*2gh*3/4 = -3mgh/4

3] acceleration = friction/m

= umg/m

= ug