A 90.3 cm long pendulum is released from a vertical height of 20 cm and is allowed to swing freely. Determine the greatest speed of the pendulum.

mgh = 1/2 mv^2 so gh = 1/2v^2

Length of the pendulum has no bearing on the question

To determine the greatest speed of the pendulum, we can use the principle of conservation of energy.

The total mechanical energy of the pendulum consists of two parts: potential energy and kinetic energy.

1. Potential Energy:
The potential energy of the pendulum at the release position (highest point) is given by the formula:
PE = mgh
where m is the mass, g is the acceleration due to gravity, and h is the vertical height.
In this case, the pendulum is released from a vertical height of 20 cm, which is equivalent to 0.2 m. Considering the mass of the pendulum is concentrated at the bob, we don't need to consider the mass in this equation.

PE = (0.2 m) * (9.8 m/s^2) * (0.2 m)
= 0.04 J

2. Kinetic Energy:
The kinetic energy of the pendulum at the lowest point is given by the formula:
KE = (1/2)mv^2
where m is the mass of the pendulum and v is the velocity.

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

0.04 J = (1/2)mv^2

However, we need to find the maximum velocity (greatest speed) of the pendulum. At this point, the kinetic energy is maximized, which is equal to its total mechanical energy.

Total energy = Potential energy + Kinetic energy
0.04 J = (1/2)mv^2
v^2 = 2 * (0.04 J)
v^2 = 0.08 J
v = √(0.08 J)
v ≈ 0.28 m/s

Therefore, the greatest speed of the pendulum is approximately 0.28 m/s.

To determine the greatest speed of the pendulum, we can use the principle of conservation of energy. The energy at the initial position (when the pendulum is released) is stored in the form of potential energy (PE) due to its height above the equilibrium position. As the pendulum swings, this potential energy is converted into kinetic energy (KE) when it reaches the lowest point of its swing.

The potential energy (PE) at the initial position can be calculated using the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height(20 cm).

PE = mgh

To calculate the kinetic energy (KE) at the lowest point, where all the potential energy is converted into kinetic energy, we use the formula KE = (1/2)mv^2, where v is the velocity.

KE = (1/2)mv^2

According to the law of conservation of energy, the total energy (TE) remains constant throughout the motion. Therefore, the potential energy at the initial position is equal to the kinetic energy at the lowest point.

PE = KE
mgh = (1/2)mv^2

We can cancel out the mass (m) from both sides of the equation:

gh = (1/2)v^2

To find the greatest speed (v), we need to solve for v. Rearranging the equation, we get:

v^2 = 2gh

Taking the square root of both sides, we have:

v = sqrt(2gh)

Substituting the given values, g = 9.8 m/s^2 and h = 20 cm = 0.2 m, we can calculate the greatest speed of the pendulum:

v = sqrt(2 * 9.8 * 0.2) m/s
v = sqrt(3.92) m/s
v ≈ 1.98 m/s

Therefore, the greatest speed of the pendulum is approximately 1.98 m/s.