A simple pendulum oscillates with an amplitude of 30 degrees. If the length of the string is 1.0m. Calculate the velocity of the pendulum bob at its lowest point.
To calculate the velocity of the pendulum bob at its lowest point, we can use the principle of conservation of energy. At the highest point of its swing, the potential energy of the pendulum bob is maximum, and at the lowest point, all of this potential energy has been converted to kinetic energy.
The potential energy of the pendulum bob at its highest point can be given as:
Potential energy (PE) = m * g * h
Where:
m = mass of the pendulum bob
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height from the lowest point to the highest point (in this case, equal to the length of the string = 1.0m)
The kinetic energy of the pendulum bob at its lowest point is:
Kinetic energy (KE) = (1/2) * m * v^2
Where:
v = velocity of the pendulum bob at the lowest point
Since the potential energy is equal to the kinetic energy, we can equate the two equations:
m * g * h = (1/2) * m * v^2
Rearranging the equation:
v^2 = 2 * g * h
Substituting the known values:
v^2 = 2 * 9.8 m/s^2 * 1.0 m
v^2 = 19.6 m^2/s^2
Taking the square root of both sides:
v ≈ 4.43 m/s
Therefore, the velocity of the pendulum bob at its lowest point is approximately 4.43 m/s.
To calculate the velocity of the pendulum bob at its lowest point, we can use the concept of conservation of energy.
At the highest point of the swing, all the potential energy is converted into kinetic energy. At the lowest point, all the potential energy is converted into kinetic energy once again.
The potential energy at the highest point is given by the formula:
Potential Energy = m * g * h
Where m is the mass of the pendulum bob, g is the acceleration due to gravity (9.8 m/s²), and h is the height measured from the lowest point to the highest point.
In this case, since we are only interested in the velocity at the lowest point, we can assume the height at the highest point is equal to the length of the string.
The potential energy at the highest point is equal to:
Potential Energy = m * g * L
Where L is the length of the string.
Now, at the lowest point, the potential energy is zero since it has all been converted to kinetic energy.
The kinetic energy of the pendulum bob at the lowest point is given by:
Kinetic Energy = (1/2) * m * v²
Where m is the mass of the pendulum bob, and v is the velocity of the pendulum bob.
Using the conservation of energy, we can equate the potential energy at the highest point to the kinetic energy at the lowest point:
Potential Energy = Kinetic Energy
m * g * L = (1/2) * m * v²
We can now solve for the velocity (v):
v = √ (2 * g * L)
Substituting the given values:
v = √ (2 * 9.8 m/s² * 1.0 m)
v = √ (19.6 m²/s²)
v ≈ 4.43 m/s
Therefore, the velocity of the pendulum bob at its lowest point is approximately 4.43 m/s.
Well, at 30 degrees, its height is
L-Lcos30
so its potential energy is mg(L-Lcos30) and that is equal to the KEnergy at the bottom...
mgL(1-cos30)=1/2 m v^2
so calculate velocity v