The problem describes a pendulum with a length $L = 2.3 \, \text{m}$ and an initial angle $\theta_0 = 38^\circ$ with the vertical. It provides information about the pendulum’s motion and asks for multiple analyses. Let’s break this down:

(a) Speed of the Bob at the Lowest Point

When the bob swings from its initial angle to the lowest point, the potential energy at the initial position is fully converted into kinetic energy at the lowest point. Using energy conservation:

$mgh_0 = \frac{1}{2}mv^2$

where:

• $m$ is the mass of the pendulum (it cancels out),
• $g$ is the acceleration due to gravity ($9.8 \, \text{m/s}^2$),
• $h_0$ is the initial height difference between the initial and lowest point,
• $v$ is the speed of the bob at the lowest point.

The height difference $h_0$ can be expressed as:

$h_0 = L(1 - \cos(\theta_0))$

Thus, the speed at the lowest point $v$ is:

$v = \sqrt{2gh_0} = \sqrt{2gL(1 - \cos(\theta_0))}$

Now substitute the values: $v = \sqrt{2 \times 9.8 \times 2.3 \times (1 - \cos(38^\circ))}$

Using $\cos(38^\circ) \approx 0.788$: $v = \sqrt{2 \times 9.8 \times 2.3 \times (1 - 0.788)} = \sqrt{2 \times 9.8 \times 2.3 \times 0.212}$ $v \approx \sqrt{9.597} \approx 3.1 \, \text{m/s}$

(b) Least Value of $v_0$ to Reach a Horizontal Position

For the bob to reach the horizontal position (where the cord is horizontal), all the kinetic energy at the lowest point must be converted into potential energy at the horizontal point. The height difference between the lowest and horizontal points is equal to the length $L$.

Using energy conservation:

$\frac{1}{2} m v_0^2 = m g L$

Cancelling $m$ and solving for $v_0$:

$v_0 = \sqrt{2gL}$

Substitute the values: $v_0 = \sqrt{2 \times 9.8 \times 2.3} = \sqrt{45.08} \approx 6.71 \, \text{m/s}$

(c) Least Value of $v_0$ to Reach the Vertical Position with the Cord Remaining Straight

In this case, for the bob to reach the vertical position, the velocity at the bottom must be large enough to create sufficient centripetal force to keep the cord taut. The condition is that the tension in the cord must at least be zero (but not negative) when the bob reaches the vertical position.

The total mechanical energy at the bottom includes both the kinetic energy at the bottom and the potential energy needed to reach the top (which is $2L$ in height):

$\frac{1}{2} m v_0^2 = m g (2L)$

Solving for $v_0$:

$v_0 = \sqrt{4gL}$

Substitute the values:

$v_0 = \sqrt{4 \times 9.8 \times 2.3} = \sqrt{90.16} \approx 9.49 \, \text{m/s}$

(d) Effect of Increasing $\theta_0$

If $\theta_0$ increases, the height difference $h_0$ increases, meaning more potential energy at the initial position, and therefore higher speeds at the lowest point. Hence, both the values for parts (b) and (c) would increase as well.

Would you like more detailed calculations or clarifications on any part?

Here are five related questions for further exploration:

1. How would the problem change if the length of the pendulum were shorter or longer?
2. What would happen to the pendulum’s speed if air resistance were included?
3. How does the pendulum's period change with different values of $\theta_0$?
4. Can we determine the time taken for the pendulum to swing from the initial position to the lowest point?
5. What happens to the tension in the string as the pendulum swings through different positions?

Tip: Always double-check your trigonometric values when solving pendulum problems to ensure accuracy in calculations.