12) (5 Points!) A block is tied to a string of length / [meters] which is pinned

Question

12) (5 Points!) A block is tied to a string of length / [meters] which is pinned down to a smooth horizontal floor. The block is pushed so that slides in uniform circular motion (where the radius of the circular trajectory is / [meters). The magnitude of the acceleration of the block is a [mls'] as it goes around the circle. Suddenly the string is cut! The block then slides straight up a nearby long, smooth ramp, inclined at an angle degrees above the horizontal. (The block's speed at the bottom of the ramp is the same speed it had when the string was cut.) Answer the following in terms of the variables given above and use g for the magnitude of free-fall acceleration. a) Find an expression for the speed of the block before the string is cut. b) Find an expression for the time it takes the block to make one complete circle before the string is cut. c) Find an expression for the distance along the ramp that the block slides before it slides back down. d) If the block hits a sticky section of the floor after sliding back down the ramp such that it slows down at a rate a [mls2] , find an expression for the distance it takes the block to come to rest in this sticky section.

Explanation / Answer

a) we know, a = v^2/l

==> v = sqrt(a*l)

b) time period, T = 2*pi*r/v

= 2*pi*l/sqrt(a*l)

= 2*pi*sqrt(l/a)

c) Apply conservation of energy

m*g*h = (1/2)*m*v^2

h = v^2/(2*g)

= a*l/(2*g)

distance travelled along the ramp, d = h/sin(theta)

= a*l/(2*g*sin(theta))

d) use, vf^2 - vi^2 = 2*a*x

0^2 - a*l = 2*(-as)*x

==> x = a*l/(2*as)

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