An object on the end of a spring with spring constant k moves in simple harmonic motion with amplitude A and frequency f. Which of the following is a possible expression for the kinetic energy of the object as a function of time t? a. kA^2sin^(2πft)
b. 1/2kA^2cos^2(2πft)
c. 1/2kA sin(2πft)
d. kAcos(2πft)
e. 1/2kAsin(2πft)

The expression for the kinetic energy of the object as a function of time, t is .

The general wave equation is given as;

Apply the principle of conservation of energy, the kinetic energy of a particle in such motion is given as;

Substitute the value of x into the kinetic energy equation

Thus, the expression for the kinetic energy of the object as a function of time, t is .

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option b.

Explanation:

The kinetic energy of a spring with constant K is calculated as:

kinetic energy = (k/2)*x^2

Where x^2 is the displacement of the spring with respect to it's rest position.

This can be written as a function like:

x = A*cos(2*pi*f*t)

where:

A is the amplitude (the maximum distance that the spring can move in each direction)

f is the frequency (and 2*pi*f is the angular frequency)

and t is the variable, it represents the time.

Replacing this in the kinetic energy equation, we get:

kinetic energy = (k/2)*(A*cos(2*pi*f*t))^2

This is the same as the option b: b. 1/2kA^2cos^2(2πft)

Then the corrrect option is b.

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