A mass of 1 kg is hanging from a spring with a spring constant of 4 N/m. At a distance of 0.2 m above the equilibrium it has a velocity of 0.8 m/s in the downward direction. What is the amplitude of the oscillation?
To find the amplitude of the oscillation, we need to use the equation for the total mechanical energy of the system.
The total mechanical energy (E) of a mass-spring system is the sum of the potential energy (PE) and the kinetic energy (KE). Mathematically, it can be expressed as:
E = PE + KE
The potential energy of the mass-spring system is given by the equation:
PE = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
The kinetic energy of the system is given by the equation:
KE = (1/2) * m * v^2
where m is the mass of the object and v is its velocity.
In this case, the mass (m) is 1 kg, the spring constant (k) is 4 N/m, the displacement (x) is 0.2 m, and the velocity (v) is 0.8 m/s.
First, let's calculate the potential energy (PE):
PE = (1/2) * k * x^2
= (1/2) * 4 N/m * (0.2 m)^2
= 0.04 J
Next, let's calculate the kinetic energy (KE):
KE = (1/2) * m * v^2
= (1/2) * 1 kg * (0.8 m/s)^2
= 0.32 J
Now we can find the total mechanical energy (E) by summing up the potential energy and the kinetic energy:
E = PE + KE
= 0.04 J + 0.32 J
= 0.36 J
The total mechanical energy (E) is constant for a mass-spring system, so it remains the same throughout the oscillation.
Now, let's consider the amplitude (A) of the oscillation. The amplitude is the maximum displacement from the equilibrium position.
At the lowest point of the oscillation, when the mass is at a distance of 0.2 m below the equilibrium, all of the mechanical energy is in the form of potential energy. Thus, the potential energy is equal to the total mechanical energy.
PE = E
0.04 J = 0.36 J
Rearranging the equation for potential energy, we can solve for the displacement (x):
PE = (1/2) * k * x^2
0.04 J = (1/2) * 4 N/m * x^2
0.02 m^2 = x^2
0.2 m = x
Therefore, the amplitude of the oscillation is 0.2 m.