A mass is oscillating with amplitude A at the end of a spring.
How far (in terms of A) is this mass from the equilibrium position of the spring when the elastic potential energy equals a half the kinetic energy?
To determine the distance of the mass from the equilibrium position when the elastic potential energy equals half the kinetic energy, we need to use the formula for the total mechanical energy of the system.
The total mechanical energy of the system comprises the sum of the kinetic energy and the elastic potential energy. At any given point in the oscillation, the total mechanical energy remains constant.
Let's denote the distance of the mass from the equilibrium position as x. The kinetic energy (KE) of the oscillating mass can be expressed as:
KE = (1/2) * m * v^2
where m is the mass of the object and v is its velocity. Since the mass is oscillating, its speed varies sinusoidally with time. At any given point, the speed can be expressed as the derivative of displacement with respect to time:
v = dx/dt
The elastic potential energy (PE) stored in the spring can be given by:
PE = (1/2) * k * x^2
where k is the spring constant.
Now, since we want to find the distance from the equilibrium position when PE is equal to half of KE, we can equate these two expressions and solve for x:
(1/2) * k * x^2 = (1/2) * m * (dx/dt)^2
To simplify the equation, let's assume that the mass of the object is denoted by m, and the spring constant is denoted by k.
Simplifying the equation further, we get:
k * x^2 = m * (dx/dt)^2
Taking the square root of both sides of the equation, we have:
x = sqrt((m/k) * (dx/dt)^2)
Now, let's integrate both sides of the equation with respect to x:
∫dx = ∫sqrt((m/k) * (dx/dt)^2)
Integrating the left side with respect to x gives us x+c_1, where c_1 is the constant of integration. Integrating the right side requires a substitution:
u = (dx/dt)
du = (d^2x/dt^2) dt
We can replace the term (dx/dt)^2 with u^2:
∫sqrt((m/k) * u^2) = ∫udt
Simplifying further:
(1/u) du = dt
Integrating the right side with respect to t gives us t+c_2, where c_2 is another constant of integration.
Thus, we have:
(1/u) du = t+c_2
To find the value of the integration constant c_2, we need additional information about the system, such as the initial conditions.