The angular velocity of the pendulum at its lowest point is approximately 3.15 rad/s.
How to find angular velocity?
When the pendulum is released from rest at an angle of 30°, it has only potential energy due to its position. The potential energy of the pendulum can be calculated using the formula:
PE = mgh
where:
PE = potential energy (in Joules)
m = mass of the pendulum (in kilograms)
g = acceleration due to gravity (9.81 m/s²)
h = height of the pendulum's center of mass above its lowest point (in meters)
The center of mass of the pendulum is located at a distance of 2/3 of the rod's length from the pivot point. The height of the center of mass at the highest point is then:
h = 2/3 × 2 m × sin(30°)
= 1.33 m
The total mass of the pendulum is 3 kg + 1 kg = 4 kg. Therefore, the potential energy at the highest point is:
PE = (4 kg) × (9.81 m/s²) × (1.33 m)
= 51.83 J
At the lowest point, the pendulum has only kinetic energy. The kinetic energy of the pendulum can be calculated using the formula:
KE = 1/2 × Iω²
where:
KE = kinetic energy (in Joules)
I = moment of inertia (in kg·m²)
ω = angular velocity (in rad/s)
The moment of inertia of a pendulum is given by:
I = m × L² + (2/5) × MR²
where:
M = mass of the bob (in kilograms)
R = radius of the bob (in meters)
In this case, M = 1 kg and R = 0.3 m. Plugging in the values:
I = 1 kg × (2 m)² + (2/5) × 1 kg × (0.3 m)²
= 4.06 kg·m²
Since there is no energy loss or gain, the potential energy at the highest point is equal to the kinetic energy at the lowest point:
PE = KE
51.83 J = 1/2 × 4.06 kg·m² × ω²
Solving for ω:
ω ≈ 3.15 rad/s
Therefore, the angular velocity of the pendulum at its lowest point is approximately 3.15 rad/s.