A 1.50 kg mass attached to a spring oscillates with a period of 0.365 s and an amplitude of 15.5 cm.
(a) Find the total mechanical energy of the system
T = 2*pi*sqrt(m/k)
so k = 4*pi^2*m/T^2 = 444.5 N/m
E = (1/2)*k*A^2 = (1/2)*444.5*0.155^2
= 5.34 J
To find the total mechanical energy of the system, we need to first determine the potential energy and kinetic energy of the oscillating mass.
The potential energy of a mass-spring system is given by the equation:
Ep = (1/2)kx^2
where:
Ep = potential energy
k = spring constant
x = displacement from equilibrium position
The kinetic energy of a mass-spring system is given by the equation:
Ek = (1/2)mv^2
where:
Ek = kinetic energy
m = mass of the object
v = velocity of the object
In the case of an oscillating mass-spring system, the potential energy is at its maximum at the extreme points of the oscillation and is zero at the equilibrium position. The kinetic energy is zero at the extreme points and at its maximum at the equilibrium position.
Given that the amplitude of the oscillation is 15.5 cm, we can determine the displacement (x) from the equilibrium position by dividing the amplitude by 2:
x = (15.5 cm) / 2 = 7.75 cm
Next, we need to determine the spring constant (k). The period (T) of oscillation and the mass (m) of the object are related to the spring constant by the equation:
T = 2π√(m/k)
Rearranging the equation, we can solve for k:
k = (4π^2m) / T^2
Substituting the given values:
m = 1.50 kg
T = 0.365 s
k = (4π^2 * 1.50 kg) / (0.365 s)^2
With the spring constant (k) determined, we can now calculate the potential energy (Ep) and kinetic energy (Ek) of the system:
Ep = (1/2)kx^2
Ek = (1/2)mv^2
Since the system oscillates, the potential and kinetic energies will be equal at any given point in time. Therefore, the total mechanical energy (E) of the system is the sum of the potential and kinetic energies:
E = Ep + Ek
By substituting the appropriate values into the equations, we can calculate the total mechanical energy of the system.