The problem involves a pendulum, where we need to find the velocity and energy changes as it moves from an initial angle. Let's break it down.

Given:

• Length of pendulum, $L = 3.0 \, \text{m}$
• Initial speed of the bob at angle $\theta_0 = 42^\circ$, $v_0 = 7.0 \, \text{m/s}$
• We assume the pendulum's bob has mass $m$ and the only forces acting are gravitational (conservative).

Questions:

(a) What is the speed at the lowest point? We will use energy conservation:

• At $\theta_0 = 42^\circ$, the pendulum has both kinetic and potential energy.
• At the lowest position, all the potential energy converts to kinetic energy.

(b) What is the least value of $v_0$ to swing up to a horizontal position?

(c) What is the least value of $v_0$ to swing to the vertical position?

(d) Will the values increase, decrease, or remain the same if $\theta_0$ is increased?

Let me start solving these, and I'll guide you through each step. Would you like the detailed breakdown of the solution?