A simple pendulum consists of a small object of mass 3.20 kg hanging at the end of a 3.70 m long light string that is connected to a pivot point. (a) Calculate the magnitude of the torque (due to the force of gravity) about this pivot point when the string makes a 7.00° angle with the vertical.

To calculate the magnitude of the torque, we can use the formula:

Torque = r * F * sin(θ)

where:
- Torque is the magnitude of the torque,
- r is the perpendicular distance between the pivot point and the line of action of force,
- F is the force,
- θ is the angle between the force vector and the line perpendicular to the lever arm.

In this case, the force is the weight of the object and can be calculated using the formula:

F = m * g

where:
- F is the force,
- m is the mass of the object,
- g is the acceleration due to gravity.

Given:
- Mass (m) = 3.20 kg
- Length of the string (r) = 3.70 m
- Angle (θ) = 7.00°
- Acceleration due to gravity (g) = 9.8 m/s² (approximate value)

Now, we can calculate the force (F) using the mass and acceleration due to gravity:

F = m * g
= 3.20 kg * 9.8 m/s²
≈ 31.36 N

Next, we need to find the perpendicular distance (r) between the pivot point and the line of action of force. In a pendulum, this distance is equal to the length of the string:

r = 3.70 m

Finally, we can substitute the values into the torque formula to calculate the magnitude of the torque:

Torque = r * F * sin(θ)
= 3.70 m * 31.36 N * sin(7.00°)

By plugging the numbers into a calculator, we find:

Torque ≈ 13.51 N·m

Therefore, the magnitude of the torque about the pivot point is approximately 13.51 N·m.