Math Snap

STEP 1

1. The gravitational potential energy change is given by $\Delta U_g = mgh$.
2. The height increase $h$ is the same for both scenarios.
3. The gravitational acceleration $g$ is constant.

STEP 2

1. Express the change in gravitational potential energy for Figure 1.
2. Express the change in gravitational potential energy for Figure 2.
3. Relate $\Delta U_{g 2}$ to $\Delta U_{g 1}$.

STEP 3

Express the change in gravitational potential energy for Figure 1:
The change in gravitational potential energy for the block in Figure 1 is:
$$\Delta U_{g 1} = mgh$$

STEP 4

Express the change in gravitational potential energy for Figure 2:
The change in gravitational potential energy for the block in Figure 2 is:
$$\Delta U_{g 2} = (2m)gh = 2mgh$$

Relate $\Delta U_{g 2}$ to $\Delta U_{g 1}$:
Since $\Delta U_{g 1} = mgh$ and $\Delta U_{g 2} = 2mgh$, we have:
$$\Delta U_{g 2} = 2 \times \Delta U_{g 1}$$ The relationship between $\Delta U_{g 2}$ and $\Delta U_{g 1}$ is:
$$\boxed{\Delta U_{g 2} = 2 \Delta U_{g 1}}$$