The figure below shows 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 θ and released.

(a) What is the speed of the ball at the lowest point if L = 2.40 m, θ = 18.0°, and m = 500 kg?
m/s

Try using conservation of energy

To find the speed of the ball at the lowest point, we can use the conservation of mechanical energy. At the highest point, the ball has gravitational potential energy (mgh) and no kinetic energy. At the lowest point, the ball has kinetic energy (1/2mv^2) and no potential energy.

Let's break down the solution into steps:

Step 1: Find the height (h) at the highest point.
Since the rod is released from an angle of 18.0°, the height at the highest point can be found using trigonometry. The height (h) is given by:

h = L * sin(θ)

Plugging in the values, we have:
h = 2.40 m * sin(18.0°)

Step 2: Find the velocity (v) at the lowest point using conservation of mechanical energy.
At the highest point, the ball has gravitational potential energy (mgh). At the lowest point, all of this potential energy is converted into kinetic energy (1/2mv^2).

mgh = 1/2mv^2

Simplifying the equation, we get:
gh = 1/2v^2
2gh = v^2

Step 3: Solve for the velocity (v) at the lowest point.
Plugging in the values, we have:
2 * 9.8 m/s^2 * h = v^2

Since we know the height (h) from Step 1, we can solve for v:

v = sqrt(2 * 9.8 m/s^2 * h)

Substitute the value of h from Step 1 to find v.

Step 4: Calculate the final answer.
Substitute the value for v into the equation to find the speed of the ball at the lowest point:

v = sqrt(2 * 9.8 m/s^2 * h) = sqrt(2 * 9.8 m/s^2 * 2.40 m * sin(18.0°))

Solve for v using a calculator, and you will find the speed of the ball at the lowest point in m/s.

To find the speed of the ball at the lowest point, we can use the conservation of mechanical energy. At the highest point, all of the gravitational potential energy is converted into kinetic energy at the lowest point, neglecting any energy losses due to friction or air resistance.

The gravitational potential energy at the highest point can be found using the equation:

PE = mgh

Where m is the mass of the ball, g is the acceleration due to gravity, and h is the height above the lowest point.

In this case, the height above the lowest point can be found using trigonometry. The height is given by:

h = L - L * cos(θ)

Where L is the length of the rod and θ is the angle through which the rod is pulled aside.

The total mechanical energy at the highest point is the sum of the gravitational potential energy and the kinetic energy at that point:

E = PE + KE

Since the rod is initially at rest, the initial kinetic energy is zero. Therefore, the total mechanical energy at the highest point is just the gravitational potential energy.

At the lowest point, all of the gravitational potential energy is converted into kinetic energy. Therefore, the kinetic energy at the lowest point is equal to the gravitational potential energy at the highest point.

So we have:

KE = PE = mgh

The kinetic energy can be expressed as:

KE = (1/2)mv²

The mass of the object, m, cancels out, and we can solve for v:

v = √(2gh)

Plugging in the given values:
- Length of the rod, L = 2.40 m
- Angle through which the rod is pulled, θ = 18.0°
- Mass of the ball, m = 500 kg
- Acceleration due to gravity, g = 9.8 m/s²

We can now calculate the speed, v:

v = √(2 * 9.8 * (2.40 - 2.40 * cos(18.0°)))

By evaluating this expression, we find that the speed of the ball at the lowest point is approximately 14.1 m/s.