Conservation of Energy — Question 1

Question 1

Mechanical energy converted during a spring excursion

A 1 kg block attached to a horizontal ideal spring of k=100 N/m is released from rest at extension $-0.30$ m. It next stops at +0.10 m on a rough fixed surface. Include block, spring and surface in a thermally insulated system; no external agent does work.

1. Find the mechanical-energy decrease and thermal-energy increase.

2. Find the displacement-averaged friction magnitude during the excursion.

3. If the maximum static friction is 10 N, can the block remain at this stop?

Question 1 - Solution

Mechanical energy converted during a spring excursion

(a) Compare the two zero-speed configurations.

Both endpoint speeds vanish, so the mechanical energies are purely elastic: $$E_i=50(.30)^2=4.5\ \mathrm J,\qquad E_f=50(.10)^2=.5\ \mathrm J.$$ $$\boxed{\Delta E_{\mathrm{mech}}=-4\ \mathrm J},\qquad\boxed{\Delta E_{\mathrm{th}}=+4\ \mathrm J}.$$

(b) Use the full path length.

Use the complete distance traveled between the stops: $$s=.10-(-.30)=.40\ \mathrm m,\qquad\boxed{\overline f=\Delta E_{\mathrm{th}}/s=4/.40=10\ \mathrm N}.$$

(c) Check static feasibility separately.

The spring force at the final point is leftward: $$F_s=-kx=-100(.10)=-10\ \mathrm N.$$ A rightward static force of 10 N is within the stated bound, $|f_s|\leq10$ N. The block can remain at rest at the limiting value; energy conservation alone does not establish this force condition.