A pendulum is formed of a particle of mass 100g suspended from the end of an inextensible string of negligible mass and of length L=100cm. The pendulum is shifted by an angle of 60 degrees from its equilibrium position about a horizontal axis passing through the other end of the string and released without initial velocity .The string is still stretched .Take the reference level of the gravitational potential energy as that of a horizontal plane containing the equilibrium position of the particle .Neglect friction.
a) Calculate the mechanical energy of the system S(pendulum, Earth, support).
b)Find in terms of theta (theta<60) the expressions for the kinetic and the potential energies of the system S.
To calculate the mechanical energy of the system S (the pendulum, Earth, support), we need to consider both the kinetic energy and potential energy.
a) The total mechanical energy of the system is the sum of the kinetic energy (KE) and the potential energy (PE):
Mechanical Energy (E) = KE + PE
Kinetic Energy (KE):
The kinetic energy of the particle at any given angle can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Since the pendulum is released without any initial velocity, the speed of the particle at any given angle will be maximum at the lowest point of the swing (equilibrium position) and zero at the highest point of the swing. The speed of the particle can be given as:
velocity = radius * angular velocity
Since the length of the pendulum is given as L = 100 cm, which is the radius of the swing, and the angular velocity is given as ω = 0 (as the particle is not in motion), the speed of the particle will be zero.
Therefore, the kinetic energy of the particle at any angle is zero.
Kinetic Energy (KE) = 0
Potential Energy (PE):
The potential energy of the particle is given by:
PE = mass * g * height
where mass is the mass of the particle (100 g = 0.1 kg), g is the acceleration due to gravity (9.8 m/s^2), and height is the vertical displacement from the reference level (equilibrium position).
At an angle of 60 degrees, the vertical displacement can be calculated using trigonometry. The vertical displacement (h) can be given as:
h = L - L * cos(theta)
Substituting the given values, we have:
h = (100 cm) - (100 cm) * cos(60 degrees)
= 100 - 100 * 0.5
= 100 - 50
= 50 cm = 0.5 m
Therefore, the potential energy at an angle of 60 degrees can be calculated as:
PE = (0.1 kg) * (9.8 m/s^2) * (0.5 m)
= 0.49 J
Now, we can calculate the mechanical energy:
Mechanical Energy (E) = KE + PE
= 0 + 0.49 J
= 0.49 J
b) To find the expressions for kinetic and potential energies in terms of theta (θ < 60 degrees), we can use the same formulas mentioned above:
Kinetic Energy (KE) = 0
Potential Energy (PE):
The vertical displacement (h) will now depend on the angle (θ) less than 60 degrees and can be calculated as:
h = L - L * cos(theta)
Therefore, the potential energy in terms of theta (θ) can be given as:
PE = (0.1 kg) * (9.8 m/s^2) * (L - L * cos(theta))
To summarize,
a) The mechanical energy of the system S is 0.49 J.
b) The kinetic energy is 0 J since the particle is not in motion, and the potential energy is (0.1 kg) * (9.8 m/s^2) * (L - L * cos(theta)).