Question

A 469 g block is released from rest at height h₀ above a vertical spring with spring constant k = 410 N/m and negligible mass. The block sticks to the spring and momentarily stops after compressing the spring 18.3 cm. How much work is done (a) by the block on the spring and (b) by the spring on the block? (c) What is the value of h₀? (d) If the block were released from height 4h₀ above the spring, what would be the maximum compression of the spring?

Answer 1

(a) Work is done by the block on the spring = 1/2*k*y^2 = 1/2*410*(0.183)^2 = 6.86 J

(b) Work done by the spring on the block = - Work done by the block = -6.86 J

(c) Taking y = 0 at the contact point we have U top = U bottom

therefore,

m*g*h0 = 1/2*k*y^2 - m*g*y

so h0 = (1/2*k*y^2 - m*g*y)/(m*g) = (1/2*410*(0.183)^2 - 0.469*9.8*0.183)/(0.469*9.8)

= ( 6.86 - 0.84) / 4.60 = 1.31 m

(d) Again we have, h0 = 4 * 1.31 = 5.24 m

So 1/2*k*y^2 - m*g*y - m*g*4h0

=> 1/2*410*y^2 - 0.469*9.8*y - 0.469*9.8*5.24 = 0

=> 205*y^2 - 4.6*y - 24.1 = 0

=> y = (4.6 +- sqrt(4.6^2 - 4*205*(-24.1)))/(2*205) = (4.6 +- 140.6) / 410 = -0.33 m, 0.35 m

Discarding the negative sign. We have y = 0.35 m

So, the maximum compression of the spring = 0.35 m.