Reading a Potential Energy Curve — Question 8

Question 8

Comparing equal-energy motion in wells of different width

Assume one-dimensional classical motion with $K=E-U(x)$ and $F_x=-dU/dx$, with no dissipation. Two equal-mass particles each have E=8 J. One moves in $U_A=x^2$ J, the other in $U_B=4x^2$ J, with x in meters.

1. Compare their speeds at x=0.

2. Find the positive turning coordinate in each well.

3. Compare the restoring force magnitudes at x=1 m and explain why equal bottom speeds do not mean identical motion.

Question 8 - Solution

Comparing equal-energy motion in wells of different width

(a) Use the equal potential minima.

Both wells have $U(0)=0$, giving identical bottom kinetic energies: $$K_A(0)=K_B(0)=8\ \mathrm J\Rightarrow\boxed{v_A(0)=v_B(0)=\sqrt{16\ \mathrm J/m}}.$$

(b) Solve U=E separately.

Solve the separate turning equations: $$x_A^2=8\Rightarrow\boxed{x_A=2\sqrt2\ \mathrm m},\qquad4x_B^2=8\Rightarrow\boxed{x_B=\sqrt2\ \mathrm m}.$$

(c) Compare local slopes.

The restoring slopes differ: $$F_A=-2x,\qquad F_B=-8x\Rightarrow\boxed{|F_A(1)|=2\ \mathrm N,\quad|F_B(1)|=8\ \mathrm N}.$$ The second well has half the amplitude and four times the force at this position. One common speed does not determine the full trajectory.