Q.1.Find the total mechanical energy in one time period? my sir says that it is (1/2)mw^2A^2 but according to me it should be 1/4mw^2A^2.How?
Q.2.The total mechnical energy of a spring mass system in SHM is E = (1/2)mw^2A^2.Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude A remains the same. The new mechanical energy will be?
Q1. w^2A^2 = V^2 at maximum velocity. At that time, all energy is KE, and equals
(1/2)mw^2A^2
Q2. Total energy is proportional to mass. But w is inversely proportional to sqrt(mass). Therefore E stays the same, if A is constant.
A.1. To find the total mechanical energy in one time period of a simple harmonic motion (SHM), we need to consider the potential energy and kinetic energy of the system.
In SHM, the potential energy is given by the equation U = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position.
Similarly, the kinetic energy is given by the equation K = (1/2)mv^2, where m is the mass of the object and v is the velocity.
In SHM, the maximum displacement from the equilibrium position is the amplitude, denoted by A. The maximum velocity reached by the object is given by v = wA, where w is the angular frequency.
Now, let's calculate the total mechanical energy in one time period.
During SHM, the object goes from one extreme position to the other extreme position, so the displacement is 2A.
At the extreme positions, the velocity is zero, so the kinetic energy is zero.
Therefore, the total mechanical energy E in one time period is equal to the potential energy at the extreme positions:
E = (1/2)k(2A)^2 = 2kA^2.
Now, comparing this with the given formula (1/2)mw^2A^2, we need to relate the angular frequency w and the spring constant k.
The angular frequency is related to the spring constant and mass by w = sqrt(k/m).
Dividing w by sqrt(k) on both sides, we get w/sqrt(k) = sqrt(m).
Substituting this value into the formula for E, we have:
E = (1/2)m(w/sqrt(k))^2A^2
= (1/2)mw^2A^2/k
= (1/2)mw^2A^2 / (m/s^2/m)
= (1/2)mw^2A^2 * m/m
= (1/2)mw^2A^2.
Hence, the correct formula for the total mechanical energy in one time period of SHM is indeed (1/2)mw^2A^2.
A.2. The total mechanical energy of a spring-mass system in SHM is given by E = (1/2)mw^2A^2, where m is the mass of the object, w is the angular frequency, and A is the amplitude of the motion.
Suppose the oscillating particle is replaced by another particle of double the mass (2m) while the amplitude A remains the same.
To find the new mechanical energy, we can simply substitute the new mass (2m) into the equation for total mechanical energy:
E' = (1/2)(2m)w^2A^2
= mw^2A^2.
Therefore, the new mechanical energy is mw^2A^2, which is half of the original mechanical energy.