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, we need to find the sum of kinetic and potential energies. Let's first find the potential energy.

a) Potential energy (PE):
The potential energy of an object under the influence of gravity is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

In this case, the height h of the pendulum can be calculated as follows:
h = L * (1 - cos(theta))
where L is the length of the string and theta is the angle the pendulum is shifted by.

Now, substitute the values into the equation:
PE = (0.1 kg) * (9.8 m/s^2) * (0.01 m) * (1 - cos(60 degrees))
= 0.1 * 9.8 * 0.01 * (1 - 0.5)
= 0.049 J

b) Kinetic energy (KE):
The kinetic energy of an object is given by the equation KE = (1/2)mv^2, where m is the mass and v is the velocity.

In this case, the pendulum is released without initial velocity, so its kinetic energy is initially zero.

So, KE = 0 J

The expression for kinetic energy doesn't depend on theta, as the pendulum is released without any initial velocity.

To summarize:
a) The mechanical energy of the system is given by the sum of potential and kinetic energy:
Mechanical Energy = PE + KE = 0.049 J + 0 J = 0.049 J

b) The expression for potential energy is: PE = (0.1 kg) * (9.8 m/s^2) * (0.01 m) * (1 - cos(theta))
The expression for kinetic energy is: KE = 0 J, as it is initially zero.

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