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To solve for the change in gravitational potential energy of the pendulum bob as it swings from point A to point B, we need to calculate the height change of the pendulum bob relative to the lowest point of its swing.

### Step 1: Calculate the vertical displacement of the bob

The pendulum string length is \( L = 1.5 \text{ m} \), and the angle at point A is \( \theta = 36^\circ \).

The vertical distance from point A to the lowest point B can be found using:
\[
h = L - L \cos(\theta)
\]
where:
- \( L \) is the length of the string,
- \( \theta \) is the angle from the vertical.

Substitute the values:
\[
h = 1.5 \text{ m} - 1.5 \text{ m} \times \cos(36^\circ)
\]

### Step 2: Calculate \( \cos(36^\circ) \)
\[
\cos(36^\circ) \approx 0.809
\]

### Step 3: Calculate \( h \)
\[
h = 1.5 \text{ m} - 1.5 \text{ m} \times 0.809 = 1.5 \text{ m} - 1.2135 \text{ m} \approx 0.2865 \text{ m}
\]

### Step 4: Calculate the change in gravitational potential energy
The change in gravitational potential energy (\( \Delta U \)) is given by:
\[
\Delta U = mgh
\]
where:
- \( m = 0.34 \text{ kg} \) (mass of the bob),
- \( g = 9.81 \text{ m/s}^2 \) (acceleration due to gravity),
- \( h = 0.2865 \text{ m} \) (vertical displacement).

Substitute the values:
\[
\Delta U = 0.34 \text{ kg} \times 9.81 \text{ m/s}^2 \times 0.2865 \text{ m} \approx 0.954 \text{ J}
\]

Thus, the change in gravitational potential energy of the pendulum bob-earth system is approximately 0.954 J.

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. How would the change in potential energy vary if the angle \( \theta \) were increased?
2. What is the kinetic energy of the pendulum bob at the bottom of its swing?
3. How would the change in potential energy be affected if the mass of the bob were doubled?
4. Can you determine the speed of the bob at point B using the conservation of energy principle?
5. How would the length of the string affect the change in gravitational potential energy?

**Tip:** In problems involving pendulums, it’s often helpful to use trigonometric identities to find height changes when given an angle and string length.

A pendulum bob with a mass of 0.34 kg is attached to a 1.5 m long string. As the pendulum bob swings from point A, where the angle θ = 36°, to point B at the bottom of its arc, determine the change in the gravitational potential energy of the pendulum bob-earth system.

Find the change in gravitational potential energy of a pendulum bob with a mass of 0.34 kg as it swings from an angle of 36° to the bottom of its arc. This detailed solution involves trigonometry and the gravitational potential energy equation, suitable for students in Grades 9-12.

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