Conservation of Mechanical Energy — Question 8

Question 8

When mass cancels and when it does not

Neglect friction and air resistance, take $g=10$ m/s$^2$, and treat all stated guides as fixed. Two sliders of masses m and 2m are each released from rest against the same spring, compressed by the same amount in separate trials. Each then climbs a smooth ramp. Compare this with releasing both sliders from a common height on a smooth track.

1. Find the ratio of launch speeds for the spring trials.

2. Find the ratio of maximum spring-driven rise heights.

3. For gravity-only release from the same height, compare speeds after equal descents and explain the difference.

Question 8 - Solution

When mass cancels and when it does not

(a) Use the same initial elastic energy S.

Write the common initial elastic energy as $S$: $$v_m=\sqrt{2S/m},\qquad v_{2m}=\sqrt{2S/(2m)}\Rightarrow\boxed{v_m/v_{2m}=\sqrt2}.$$

(b) Convert the same energy into mgh.

At the respective turning heights, $$h_m=\frac S{mg},\qquad h_{2m}=\frac S{2mg}\Rightarrow\boxed{h_m/h_{2m}=2}.$$

(c) Compare how the initial store scales.

For gravity-only release through the same height $H$, $$mgH=\tfrac12mv^2\Rightarrow\boxed{v_m=v_{2m}=\sqrt{2gH}}.$$ The gravitational store scales with mass and cancels the inertia factor. A fixed spring compression supplies the same energy to either mass.