The block in the figure lies on a horizontal frictionless surface, and the sprin

Question

The block in the figure lies on a horizontal frictionless surface, and the spring constant is 48 N/m. Initially, the spring is at its relaxed length and the block is stationary at position x = 0. Then an applied force with a constant magnitude of 3.0 N pulls the block in the positive direction of the x axis, stretching the spring until the block stops. When that stopping point is reached, what are (a) the position of the block, (b) the work that has been done on the block by the applied force, and (c) the work that has been done on the block by the spring force? During the block's displacement, what are (d) the block's position when its kinetic energy is maximum and (c) the value of that maximum kinetic energy?

Explanation / Answer

(a) At stopping position, net force is zero.

F - kx = 0

=> x = F/k = 3.0 / 48 = 0.0625 m

(b) Work done by applied force, W = Fx = 3.0 * 0.0625 = 0.19 J

(c) Work done by spring = Energy stored in spring = kx2/2 = 48 * 0.06252 / 2 = 0.094 J

(d) Maximum KE means means maximum velocity.

Net force, Fnet = F - kx

=> mv(dv/dx) = F - kx

=> mvdv = (3 - 48x)dx

Integraing both sides,

mv2/2 = 3x - 24x2 + c

=> Kinetic energy, K = 3x - 24x2 + c

At K = Kmax, dK/dx = 0

=> 3 - 48x = 0

=> x = 0.0625 m

(e) Using the work energy theorem, KE is given by,

kx2/2 + K= Fx

=> K= Fx - kx2/2

=> Kmax = (3.0 * 0.0625) - (48 * 0.06252 / 2) = 0.094 J

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