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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?

The figure shows a pendulum of length L = 3.0 m. Its bob (which effectively has all the mass) has speed v0 when the cord makes an angle θ0 = 42° with the vertical. (a) What is the speed of the bob when it is in its lowest position if v0 = 7.0 m/s? What is the least value that v0 can have if the pendulum is to swing down and then up (b) to a horizontal position, and (c) to a vertical position with the cord remaining straight? (d) Do the answers to (b) and (c) increase, decrease, or remain the same if θ0 is increased by a few degrees?

Learn how to calculate the speed of a pendulum's bob at its lowest position and the minimum initial speed required for the bob to reach horizontal and vertical positions. This problem involves applying the law of energy conservation and kinematics. Suitable for high school and early college physics students.

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