Question

A mass m = 16 kg rests on a frictionless table and accelerated by a spring with spring constant k = 4739 N/m. The floor is frictionless except for a rough patch. For this rough path, the coefficient of friction is μ_k = 0.45. The mass leaves the spring at a speed v = 2.9 m/s.

1)How much work is done by the spring as it accelerates the mass?

2)How far was the spring stretched from its unstreched length?

3)The mass is measured to leave the rough spot with a final speed v_f = 1.7 m/s. How much work is done by friction as the mass crosses the rough spot?

4)What is the length of the rough spot?

5)In a new scenario, the block only makes it (exactly) half-way through the rough spot. How far was the spring compressed from its unstretched length?

6)in this new scenario, what would the coefficient of friction of the rough patch need to be changed to in order for the block to just barely make it through the rough patch?

Answer 1

1) The work done by the spring is converted into kinetic energy of the body as there is no friction. The kinetic energy of the block when it leaves the spring is 1/2 m v2

W = 1/2 x 16 x 2.92 = 67.28 J

2) Intial elastic energy stored in the spring is converted into kinetic energy of the block, Hence

1/2 k x2 = 1/2 m v2 =>  x2 = m v2 / k = 16 x 2.92 / 4739 = 0.0284; then x = 0.17 m

The compression in the spring is 0.17m

3) The speed of the body before rough spot is 2.9 m/s and its speed after rough spot is 1.7 m/s. The decrease in kinetic energy is due to workdone by frictional force.

Hence Wf = KEf - KEi = 1/2 m Vf2 - 1/2 m Vi2 = 1/2 m (Vf2 - Vi2) = 1/2 x 16 x (1.72 - 2.92)

The work done by friction is Wf = - 44.16 J

4) Wf = mg x S   => 44.16 = 0.45 x 16 x 9.8 x S => S = 0.626 m = 62.6 cm

The length of the rough patch is 0.626m

5) If the body travels only half of the rough patch then the work done by the spring is equal to work done against the frictional force. Let y is new compression in spring.

1/2 k y2 = mg x S/2 =>  y2 = mg x S / k = 0.45 x 16 x 9.8 x 0.626 / 4739 = 0.0093

The new compression in the spring is y = 0.096 m = 9.6 cm

6) The body crosses the rough patch in this new scenario if the coefficient of kinetic friction is half of its original value. The new coefficient of kinetic friction is 0.225.