Question

A pendulum is made by letting a 4 kg mass swing at the end of a string that has a length of 1.5 meter. The maximum angle that the string makes with the vertical is 30 degrees (in both directions). What is the change in the potential energy of the mass as it moves from the lowest point to the highest point of its path?

Answer 1

Answer:

Approximately $7.8\; \rm J$.

Explanation:

The change in the gravitational potential energy of the pendulum is directly related to the change in its height.

Refer to the sketch attached. The pendulum is initially at $\rm P_2$. Its highest point is at $P_1$. The length of segment $\rm BP_2$ gives the change in its height.

The lengths of $\rm AP_1$ and $\rm AP_2$ are simply the length of the string, $1.5\; \rm m$. To find the length of $\rm BP_2$, start by calculating the length of $\rm AB$.

$\rm AB$ forms a leg in the right triangle $\rm \triangle AP_1B$. Besides, it is adjacent to the $30^\circ$ angle $\rm P_1\hat{A}B$. Its length would be:

$\rm AB = 1.5 \times \cos(30^\circ) \approx 1.30\; \rm m$.

The length of $\rm BP_2$ would thus be

$\rm BP_2 = AP_2 - AB = 1.5 - 1.30 \approx 0.20\; \rm m$.

The change in gravitational potential energy can be found with the equation

$\Delta \mathrm{GPE} = m \cdot g \cdot \Delta h$. In this equation,

• $m$ is the mass of the object,
• $g \approx 9.81\; \rm N \cdot kg^{-1}$ near the surface of the earth, and
• $\Delta h$ is the change in the object's height.

In this case, $m = 4\; \rm kg$ and $\Delta h \approx 0.20\; \rm m$. Therefore:

$\Delta \mathrm{GPE} = 4 \times 9.81 \times 0.20 \approx 7.8\; \rm J$.