A pendulum bob has a mass of 1 kg. The length of the pendulum is 2 m. The bob is pulled to one side to an angle of 10 degrees from the vertical.
What is the velocity of the pendulum bob as it swings through its lowest point?
What is the angular velocity?
the bob's max height is 0.03m
so the KE at the bottom is the same as the PE at the top of the swing.
1/2 v^2 = .03*9.81
now, ω = v/r
answer
Yes
LLL
To find the velocity of the pendulum bob as it swings through its lowest point, we can use the principle of conservation of mechanical energy. The mechanical energy of the pendulum is conserved, which means it remains constant throughout its motion.
The mechanical energy of the pendulum is the sum of its kinetic energy (KE) and potential energy (PE). At the highest point of its swing, when the bob is pulled to one side, it only has potential energy due to its height. At the lowest point of its swing, all the potential energy is converted into kinetic energy, resulting in maximum velocity.
To calculate the velocity, we need to find the potential energy at the maximum height and equate it to the kinetic energy at the lowest point.
The potential energy (PE) is given by the formula: PE = m * g * h
Where:
m = mass of the pendulum bob (1 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height above the lowest point at the highest point of the swing
The height can be calculated using trigonometry. The maximum height reached by the bob is given by: h = L * (1 - cos(theta))
Where:
L = length of the pendulum (2 m)
theta = angle from the vertical (10 degrees)
Substituting the values into the equations, we have:
h = 2 * (1 - cos(10°)) ≈ 0.0383 m
Now we can calculate the potential energy:
PE = 1 kg * 9.8 m/s^2 * 0.0383 m ≈ 0.377 J
Since mechanical energy is conserved, the kinetic energy at the lowest point will be equal to the potential energy at the highest point. So, the kinetic energy (KE) will also be 0.377 J.
The kinetic energy can be given by the formula: KE = (1/2) * m * v^2
Rearranging the equation, we can find the velocity (v):
v = sqrt((2 * KE) / m)
Substituting the values:
v = sqrt((2 * 0.377 J) / 1 kg) ≈ 0.775 m/s
Therefore, the velocity of the pendulum bob as it swings through its lowest point is approximately 0.775 m/s.
Now, let's calculate the angular velocity of the pendulum.
Angular velocity is the rate at which the pendulum bob rotates or swings. It is usually represented by the symbol ω (omega) and is measured in radians per second.
The formula for angular velocity is given by: ω = v / r
Where:
v = linear velocity (0.775 m/s, which we calculated earlier)
r = radius of the circular path (which is equal to the length of the pendulum, L)
Substituting the values:
ω = 0.775 m/s / 2 m ≈ 0.3875 rad/s
Therefore, the angular velocity of the pendulum is approximately 0.3875 rad/s.