A simple pendulum of length 1 metre has a bob of mass 100 grams.It is displaced through an angle 60 degree and then released.Calculate the speed of the bob when the pendulum passes through the mean position.
To calculate the speed of the bob when the pendulum passes through the mean position, we can use the principles of conservation of mechanical energy.
First, let's find the gravitational potential energy at the maximum displacement. The equation for gravitational potential energy is:
Potential Energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Since the pendulum is displaced through an angle of 60 degrees, the height at the maximum displacement is the vertical component of the displacement, which is given by:
h = length (L) * sin(angle)
Plugging in the values, we have:
h = 1 m * sin(60°)
Next, we'll calculate the gravitational potential energy at the mean position (where the pendulum passes) where h = 0. The potential energy in this position will be:
PE = m * g * h
Since h = 0, the potential energy at the mean position is zero.
According to the principle of conservation of mechanical energy, the total mechanical energy of the system remains constant. Therefore, the initial mechanical energy at the maximum displacement is equal to the mechanical energy at the mean position.
The mechanical energy of a pendulum is the sum of its kinetic energy and potential energy. At maximum displacement, the entire energy is in the form of potential energy. At the mean position, all the energy is in the form of kinetic energy.
Therefore, we have:
Potential Energy at maximum displacement = Kinetic Energy at mean position
Using the equations for potential and kinetic energy, we can calculate the speed (v) at the mean position:
Potential Energy at maximum displacement = (1/2) * m * v^2
Solving for v, we get:
v = sqrt(2 * PE / m)
Since the potential energy at the maximum displacement is equal to the mass times the gravitational potential energy at the maximum displacement, we have:
v = sqrt(2 * (m * g * h) / m)
Plugging in the values:
v = sqrt(2 * (0.1 kg * 9.8 m/s^2 * (1 m * sin(60°))) / 0.1 kg)
v ≈ sqrt(2 * (9.8 m^2/s^2 * 0.866 m) / 1)
v ≈ sqrt(16.944) m/s
v ≈ 4.12 m/s
Therefore, the speed of the bob when the pendulum passes through the mean position is approximately 4.12 m/s.