This problem is intended to be solved only using variables, no numbers. A block

Question

This problem is intended to be solved only using variables, no numbers.

A block of mass, 'm' is placed against a spring of constant, 'k' that is compressed a distance,' x' from its equilibrium position, (see below) When the spring is released, the block slides across the frictionless surface and then goes up a ramp that has friction with coefficients mu k and mu s. Starting with W AE, derive an expression for how high the block will go. If the block finally comes to rest on the ramp, what must be true about mu s? (Start your derivation with a Free Body Diagram)

Explanation / Answer

(a) The energy imparted to the block due to spring compression = 0.5k (x)^2. Since there is no friction, the only loss is due to work done against gravity and friction.

Thus, this energy should be equal to mgh where h is verticle distance from ground + work done against friction.

work done against friction = friction force x distance = (k mgcos) (h/sin) = kmgh/tan.

Thus, 0.5k (x)^2 = kmgh/tan + mgh = h [k mg/tan + mg].

Thus, h = 0.5k (x)^2 / [k mg/tan + mg].

(b) When the block comes to rest, there is still a force mg sin pulling it down the ramp. But, the friction stops it from moving. the friction force is k mgcos which must be GREATER than the mgsin which is why it "wins" and stops the block from moving. Thus, k mgcos > mgsin and hence, k > tan.

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