A 4.0 kg pendulum bob swings on a 2-m-long string. The bob's speed is 1.5 m/s when the string makes a 15 ∘ angle with vertical and the bob is moving toward the bottom of the arc. What is the magnitude of the bob's tangential acceleration at this instant?

The tangential acceleration of the pendulum bob is due to the component of gravitational force acting along the direction of the bob’s motion at that instant.

To find the tangential acceleration (\( a_t \)), we use the following relationship:

\[
a_t = g \sin(\theta)
\]

Where:
- \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)),
- \( \theta \) is the angle with the vertical (\( 15^\circ \)).

Now, let’s calculate the tangential acceleration:

\[
a_t = 9.8 \times \sin(15^\circ)
\]

We first need to compute \( \sin(15^\circ) \):

\[
\sin(15^\circ) \approx 0.2588
\]

Now calculate \( a_t \):

\[
a_t = 9.8 \times 0.2588 \approx 2.54 \, \text{m/s}^2
\]

Thus, the magnitude of the bob’s tangential acceleration at that instant is approximately:

\[
a_t \approx 2.54 \, \text{m/s}^2
\]

Would you like further details on this, or have any questions?

Here are five related questions to expand on this:
1. How does the mass of the pendulum affect the tangential acceleration?
2. What is the relationship between tangential and centripetal accelerations in pendulum motion?
3. How would the tangential acceleration change if the angle increased?
4. What is the velocity of the bob at the lowest point of the swing?
5. How is the total mechanical energy conserved in pendulum motion?

**Tip:** The tangential acceleration depends only on the gravitational force and the angle, not on the mass of the pendulum bob.

Learn how to calculate the tangential acceleration of a 4.0 kg pendulum bob on a 2-meter string swinging at a 15-degree angle with a speed of 1.5 m/s. This problem covers key physics concepts, including pendulum motion, gravitational forces, and the use of trigonometric functions. Suitable for high school students in Grades 11-12.

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