A thin rod, of length L and negligible mass, that can pivot about one end to rotate in a vertical circle. A heavy ball of mass m is attached to the other end. The rod is pulled aside through an angle and released.

(a)What is the speed of the ball at the lowest point if L = 2.20 m, = 19.0°, and m = 500 kg?
(b) Does the speed increase, decrease, or remain the same if the mass is increased?

Use energy techniques.

I will be happy to critique your thinking on this.

To solve this problem using energy techniques, we can analyze the conservation of mechanical energy of the system.

Let's start by finding the initial potential energy of the system. The gravitational potential energy at the initial angle is given by:

PE_initial = m * g * h_initial

where m is the mass of the ball, g is the acceleration due to gravity, and h_initial is the initial height of the ball.

In this case, the initial height of the ball can be found using basic trigonometry. Since the rod length and the initial angle are given, we can calculate the height using the equation:

h_initial = L * (1 - cos(θ_initial))

where L is the length of the rod and θ_initial is the initial angle.

Next, let's find the potential energy of the system at the lowest point. At the lowest point, the ball is at its maximum height, which is given by:

h_lowest = L * (1 - cos(180°))

Using the same formula as before, we can find the gravitational potential energy at the lowest point:

PE_lowest = m * g * h_lowest

The kinetic energy at the lowest point is given by:

KE_lowest = (1/2) * m * v_lowest^2

where v_lowest is the velocity of the ball at the lowest point.

Using the principle of conservation of mechanical energy, we can equate the initial potential energy to the sum of the kinetic and potential energy at the lowest point:

PE_initial = PE_lowest + KE_lowest

Let's rearrange the equation to solve for v_lowest:

v_lowest = sqrt(2 * (PE_initial - PE_lowest) / m)

Substituting the formulas for potential energy and simplifying the equation:

v_lowest = sqrt(2 * g * L * (cos(θ_initial) - cos(180°)))

Now we can plug in the given values to calculate the speed of the ball at the lowest point.

(a) Given: L = 2.20 m, θ_initial = 19.0°, m = 500 kg

Using the formula derived earlier:

v_lowest = sqrt(2 * 9.8 m/s^2 * 2.20 m * (cos(19.0°) - cos(180°)))

Solving this equation will give us the speed of the ball at the lowest point.

(b) If the mass is increased, the speed at the lowest point would remain the same. This is because the speed at the lowest point depends on the initial potential energy and the geometry of the system, but it is independent of the mass. So increasing the mass will not affect the speed at the lowest point.