Work Done on a System by an External Force — Question 9

Question 9

A variable external force against a spring and friction

Assume no heat transfer across the stated system boundary during the process unless otherwise specified. A 2 kg block starts at rest at x=0, where its horizontal spring is relaxed. An external force $F_a=10+2x$ N pulls it right to x=2 m. The spring has k=4 N/m; limiting static friction and sliding friction are both 1 N. Include spring and floor in the system.

1. Find external work, spring-energy increase and thermal increase.

2. Find final speed.

3. Check that the block can move continuously rightward throughout the stated interval.

Question 9 - Solution

A variable external force against a spring and friction

(a) Integrate the changing applied force.

Integrate the applied force and evaluate the internal changes: $$\boxed{W_a=\int_0^2(10+2x)dx=[10x+x^2]_0^2=24\ \mathrm J},$$ $$\boxed{\Delta U_s=\tfrac12(4)(2)^2=8\ \mathrm J},\qquad\boxed{\Delta E_{\mathrm{th}}=1(2)=2\ \mathrm J}.$$

(b) Use the energy left for motion.

The remaining input becomes kinetic energy: $$K_f=24-8-2=14\ \mathrm J\Rightarrow\boxed{v_f=\sqrt{2(14)/2}=\sqrt{14}\ \mathrm{m/s}}.$$

(c) Check intermediate states and the start.

The applied initial 10 N exceeds the 1 N static threshold. During sliding, $$K(x)=10x+x^2-2x^2-x=9x-x^2>0\quad(0<x\leq2),$$ $$F_{\mathrm{net}}=10+2x-4x-1=9-2x\geq5\ \mathrm N.$$ The block starts and continues rightward throughout the required interval.