A 7.0-kg block on a horizontal frictionless surface is attached to a light spring (force constant = 1.2 kN/m).The block is initially at rest at its equilibrium position when a force of magnitude P acting parallel to the surface is applied to the block,as shown.When the block is 8.0 cm from the equilibrium position,it has a speed of 0.80 m/s.How much work is done on the block by the force P as the block moves the 8.0 cm?
A)7.4 J
B)5.4 J
C)6.1 J
D)6.7 J
E)4.9 J

Question

A 7.0-kg block on a horizontal frictionless surface is attached to a light spring (force constant = 1.2 kN/m).The block is initially at rest at its equilibrium position when a force of magnitude P acting parallel to the surface is applied to the block,as shown.When the block is 8.0 cm from the equilibrium position,it has a speed of 0.80 m/s.How much work is done on the block by the force P as the block moves the 8.0 cm? 
A)7.4 J 
B)5.4 J 
C)6.1 J 
D)6.7 J 
E)4.9 J

Quizplus · Accepted Answer

The potential energy of the spring is given by:
U_spring = (1/2)kx^2
where k is the spring constant and x is the displacement from the equilibrium position.
At the equilibrium position, the potential energy of the spring is zero. So, at a displacement of 8.0 cm from the equilibrium position, the potential energy of the spring is:
U_spring = (1/2)(1.2 kN/m)(0.08 m)^2 = 0.0384 J
The total mechanical energy (kinetic plus potential) of the block-spring system is conserved because there is no friction. Therefore, the work done by the force P is equal to the increase in mechanical energy of the system.
The initial mechanical energy of the system is:
E_i = (1/2)mv_i^2 + U_spring = (1/2)(7.0 kg)(0.80 m/s)^2 + 0.0384 J = 2.816 J
The final mechanical energy of the system is:
E_f = (1/2)mv_f^2 + (1/2)kx_f^2 = (1/2)(7.0 kg)(0 m/s)^2 + (1/2)(1.2 kN/m)(0.08 m)^2 = 0.0384 J
The work done by the force P is:
W = E_f - E_i = 0.0384 J - 2.816 J = -2.7776 J
The negative sign indicates that the work done by the force P is negative (i.e., the force opposes the displacement of the block). Therefore, the correct answer is:
Choice C) 6.1 J

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