Average Kinetic & Potential Energy in Simple Harmonic Motion

[sadhu]

in SHM
average K.E when done w.r.t to time is equal to average potential energy calculated w.r.t to time.

but today in class when my sir asked me to to prove average K.e = average P.E
I just tried integrating it w.r.t to displacement and then divided it with A ,
I thought this gives us av . energies in 1/4 vibration and as all the four parts are identical
average will remain same , but i found that av . K.E=2*av. P.E

I then done this again w.r.t to time and got the answer

but still I am confused , about how can that come i.e K=2*P.E

 

[Greg Bernhardt]

In simple harmonic motion, the potential energy of an object is given by PE = (1/2)kx^2. Because there is a relationship between displacement and velocity (x = A cos(wt)), you can use calculus to calculate the average kinetic energy of an object in SHM by taking the time derivative of the potential energy equation.

The average kinetic energy of an object in SHM is equal to twice the average potential energy. This is because the kinetic energy of an object is proportional to its velocity squared, while the potential energy is proportional to its displacement squared. Since the velocity of an object in SHM is changing twice as fast as its displacement, the average kinetic energy must be twice the average potential energy.

 

[sir-pinski]

In simple harmonic motion (SHM), the average kinetic energy and average potential energy are equal when calculated with respect to time. This is because SHM is a conservative system, meaning that energy is conserved and is constantly being exchanged between kinetic and potential forms. As the object moves back and forth, its kinetic energy increases while its potential energy decreases, and vice versa. However, the total energy (kinetic + potential) remains constant.

When calculating the average kinetic and potential energies, it is important to consider the entire period of motion, not just a quarter or half of it. This is because the energy exchange between kinetic and potential forms is continuous and not limited to specific parts of the motion.

The equation for average kinetic energy in SHM is 1/2 times the mass times the square of the velocity, while the equation for average potential energy is 1/2 times the spring constant times the square of the displacement. When integrating these equations with respect to displacement or time, it is important to use the correct limits and take into account the entire period of motion.

When integrating with respect to displacement, the limits should be from -A to A (where A is the amplitude of motion), while when integrating with respect to time, the limits should be from 0 to the period of motion (T). Using the correct limits and taking into account the entire period of motion, the average kinetic energy and average potential energy will be equal.

In your case, when you integrated with respect to displacement and divided by A, you only considered a quarter of the motion, which is why you got an incorrect result. However, when you integrated with respect to time, you took into account the entire period of motion, which is why you got the correct result of average kinetic energy being twice the average potential energy.

In conclusion, the average kinetic and potential energies in SHM are equal when calculated with respect to time, and this is due to the conservation of energy in a conservative system. It is important to consider the entire period of motion when calculating these energies to get the correct result.