A clever engineer designs a \"sprong\". When laying on the x-axis it obeys the f

Question

A clever engineer designs a "sprong". When laying on the x-axis it obeys the force law F = (-q(x - x_c)^3, 0, 0), where x_c is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, we'll let x_c = 0m. Then F = (-qx^3, 0, 0). a. What are the units of q? b. Find an expression for the potential energy of a stretched or compressed sprong. c. The sprong from above is used in a sprong-loaded toy gun that shoots a 20 g plastic ball. How much does the mass of the system increase if the sprong constant is 40, 000, with the units you found in part a, and the sprong is compressed 10 cm? d. What is the launch speed if the sprong is compressed 10 cm? Assume the barrel is frictionless. e. Draw the K, U, and K + U plot for the system.

Explanation / Answer

a) F = N , and units of ( x-xc)^3 = m^3

N = q ( m^3)

q= N/m^3 ( units)

b) F = dU/fx

dU = Fdx = -q x^3 dx

U = -q x^4/4

c) mg = qx^3

m (9.8) = 40,000 (N/m^4) (0.1)^3

m = 4.081 kg

d) -qx^4/4 = 1/2mv^2

(40000) ( 0.1)^4/ 4 = 1/2 ( 20 x 10^-3 ) v^2

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