An object with mass 0.250 kg is acted on by an elastic restoring force with force constant 10.8 N/m. The object is set into oscillation with an initial potential energy of 0.150 J and an initial kinetic energy of 6.00×10−2 J.
What is the amplitude of oscillation?
What is the potential energy when the displacement is one-half the amplitude?
At what displacement are the kinetic and potential energies equal?
What is the value of the phase angle \phi if the initial velocity is positive and the initial displacement is negative?
To find the amplitude of oscillation, we can use the formula for the total mechanical energy of a system undergoing simple harmonic motion:
Total mechanical energy (E) = Kinetic energy (K) + Potential energy (U)
Given that the initial potential energy (U) is 0.150 J and the initial kinetic energy (K) is 6.00×10^(-2) J, we can substitute these values into the equation and solve for the total mechanical energy:
E = K + U
E = 6.00×10^(-2) J + 0.150 J
E = 0.210 J
The total mechanical energy (E) of the system is constant and equal to the sum of kinetic and potential energies. In simple harmonic motion, the maximum potential energy is equal to the maximum kinetic energy.
So, we can take the maximum kinetic energy to be half of the total mechanical energy:
K_max = 0.5 * E
K_max = 0.5 * 0.210 J
K_max = 0.105 J
The maximum potential energy is also equal to half of the total mechanical energy:
U_max = 0.5 * E
U_max = 0.5 * 0.210 J
U_max = 0.105 J
At the displacement of one-half the amplitude, the potential energy (U) is equal to:
U = 0.5 * U_max
U = 0.5 * 0.105 J
U = 0.0525 J
To find the displacement at which the kinetic and potential energies are equal, we can set the expressions for kinetic energy (K) and potential energy (U) equal to each other:
K = U
Since we have already found that the maximum potential energy and maximum kinetic energy are both equal to 0.105 J, we can write:
0.5 * m * v_max^2 = 0.5 * k * x_max^2
The mass (m) and force constant (k) are given as 0.250 kg and 10.8 N/m, respectively. We can substitute these values and solve for the displacement (x_max):
0.5 * (0.250 kg) * v_max^2 = 0.5 * (10.8 N/m) * x_max^2
Simplifying the equation gives:
v_max^2 = (10.8 N/m) * (x_max^2) / (0.250 kg)
To find the displacement at which the kinetic and potential energies are equal, we need to find the velocity at that point. We can use the conservation of mechanical energy to relate the kinetic and potential energies at any given displacement.
Since the total mechanical energy (E) is constant, we have:
K + U = E
Substituting the expressions for kinetic and potential energies, we get:
0.5 * m * v^2 + 0.5 * k * x^2 = E
In this equation, we know the values for all the variables (mass m, force constant k, and the total mechanical energy E). We can rearrange the equation to solve for the velocity (v) at any given displacement (x).
To find the value of the phase angle (φ), we need more information about the system. The phase angle is related to the initial conditions of the system, such as the initial displacement (x) and initial velocity (v).
Without further information, we cannot determine the exact value of the phase angle (φ) if the initial velocity is positive and the initial displacement is negative.