QuestionFigure 1
Figure 2
In Figure 1, a block of mass $m$ has speed $v$ when it reaches the bottom of a ramp. When the block reaches the top of the ramp, the height of the block has increased by $h$ and the gravitational potential energy of the block-Earth system has changed by an amount $\Delta U_{g 1}$ . In Figure 2 , a block of mass $2 m$ has speed $v / 2$ when it reaches the bottom of the same ramp. When the block reaches the top of the ramp, the height of the block increases by $h$ and the gravitational potential energy of the block-Earth system changes by an amount $\Delta U_{g 2}$ . Which of the following correctly relates $\Delta U_{g 2}$ to $\Delta U_{g 1}$ ?

Studdy Solution

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:
$\Delta U_{g 2} = (2m)gh = 2mgh$

STEP 5

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:
$\boxed{\Delta U_{g 2} = 2 \Delta U_{g 1}}$

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