A 1.2 kg mass attached to a spring oscillates with an amplitude of 5.1 cm
and a frequency of 1.2 Hz. What is its energy of motion?
y = 0.051 sin (2 pi *1.2 t ) = 0.051 sin (7.74 t )
dy/dt = velocity = (0.051 * 7.74) cos (7.74 t)
all kinetic energy when cos wt = 1, no potential E then because spring is not stretched (sin w t = 0)
Ke max = (1/2) m vmax^2 = (1/3)(1.2) ( 0.051*7.74)^2 Joules
To find the energy of motion of the spring mass system, we can use the formula for the energy of simple harmonic motion:
E = 1/2 * k * A^2
where E is the energy of motion, k is the spring constant, and A is the amplitude of the motion.
To find the spring constant, we need to use the formula for the angular frequency of the system:
ω = 2πf
where ω is the angular frequency and f is the frequency of the oscillation.
Given that the frequency is 1.2 Hz, the angular frequency is:
ω = 2π * 1.2 = 7.536 rad/s
Now, we can find the spring constant using the formula:
k = mω^2
where m is the mass attached to the spring.
Given that the mass, m, is 1.2 kg, we can calculate the spring constant:
k = 1.2 kg * (7.536 rad/s)^2 = 68.04 N/m
Finally, we can substitute the values of the spring constant (k = 68.04 N/m) and the amplitude (A = 0.051 m) into the energy formula:
E = 1/2 * 68.04 N/m * (0.051 m)^2
Simplifying the equation, we find:
E ≈ 0.087 J
Therefore, the energy of motion of the spring-mass system is approximately 0.087 Joules.
To find the energy of motion of the mass attached to a spring, we can use the formula for the energy of motion in simple harmonic oscillators, given by:
E = (1/2) * m * v^2
where E is the energy of motion, m is the mass, and v is the velocity.
First, let's find the velocity of the mass at its maximum displacement (amplitude).
The maximum displacement (A) is given as 5.1 cm, or 0.051 m.
The formula for the velocity of a mass in simple harmonic motion is given by:
v = 2πfA
where f is the frequency and A is the amplitude.
Substituting the given values, we have:
v = 2π * 1.2 Hz * 0.051 m
v ≈ 0.39 m/s
Now we can calculate the energy of motion using the formula:
E = (1/2) * m * v^2
Given that the mass (m) is 1.2 kg, and the velocity (v) is 0.39 m/s, we can substitute these values into the formula:
E = (1/2) * 1.2 kg * (0.39 m/s)^2
E ≈ 0.091 J
Therefore, the energy of motion of the mass attached to the spring is approximately 0.091 Joules.