A 0.20 kg mass and on a horizontal frictionless surface is attached to one end o

Question

A 0.20 kg mass and on a horizontal frictionless surface is attached to one end of a horizontal spring (with k = 550 N/m) whose other end is fixed. The mass has a kinetic energy of 12.0 as it passes through its equilibrium position (the point at which the spring force is zero). At what rate is the spring doing work on the mass as the mass passes through its equilibrium position? At what rate is the spring doing work on the mass when the spring is compressed 0.104 m and the mass is moving away from the equilibrium position?

Explanation / Answer

Let:
v be the velocity,
x be the displacement,
w be the angular frequency,
a be the amplitude,
k be the spring constant,
K be the kinetic energy at displacement x,
K0 be the kinetic energy at the equilibrium position,
P be the required power.

w^2 = k / m
v^2 = w^2(a^2 - x^2)
= k(a^2 - x^2) / m
v = sqrt[ k(a^2 - x^2) / m ] ...(1)

K = mv^2 / 2
= k(a^2 - x^2) / 2 ...(2)

Putting x = 0 and K = K0:
K0 = k a^2 / 2 ...(3)
a^2 = 2K0 / k ...(4)

Eliminating a^2 from (2) and (3):
K = k(2K0 / k - x^2) / 2
= K0 - kx^2 / 2

P = dK / dt = - kx dx / dt
= - kxv

Substituting for v from (1):
P = - kx sqrt[ k(a^2 - x^2) / m ]

Substituting for a^2 from (4):
P = - kx sqrt[ k(2K0 / k - x^2) / m ]
= - kx sqrt[ (2K0 - kx^2) / m ]
= - 550 * 0.104 sqrt[ (2 * 12.0 - 550 * 0.104^2) / 0.20 ]
= - 543.41 watt .

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