A mass of 0.3 kg is attached to the end of a massless spring of spring constant 0.25 N/m. It is released from rest (t=0). After 0.7 s, the speed of the mass is measured to be 2.0 m/s. What is the amplitude of oscillation?
To find the amplitude of the oscillation, we can use the equation for the displacement of a mass-spring system:
x(t) = A * cos(ωt + φ)
Where:
- x(t) is the displacement of the mass at time t,
- A is the amplitude of the oscillation,
- ω is the angular frequency, and
- φ is the phase constant.
To find A, we need to find the value of x(t) at a given time (in this case, t = 0.7 s) and calculate the displacement. The problem states that the speed of the mass at that time is 2.0 m/s.
Speed is the magnitude of velocity, and in simple harmonic motion, the magnitude of velocity is given by:
v(t) = ωA * sin(ωt + φ)
Since the mass is at its maximum displacement (the amplitude) when the speed is zero, we can set v(t) = 0 to find that point.
0 = ωA * sin(ωt + φ)
sin(ωt + φ) = 0
This means that ωt + φ is equal to multiples of π, as sin(θ) = 0 for θ = nπ, where n is an integer.
ωt + φ = nπ
Since we are specifically interested in the displacement at t = 0.7 s, we know that ωt = ω(0.7) = π/2, since ω = 2πf and f = 1/T, where T is the period.
π/2 + φ = nπ
From this equation, we can determine the phase constant φ. Notice that φ can be any angle in which π/2 is greater than. So, let's assume φ = 0.
π/2 + 0 = nπ
π/2 = nπ
If we solve for n, we get n = 1/2.
Now that we have the phase constant φ, we can find A by finding the displacement x(t) at t = 0.7 s.
x(t) = A * cos(ωt + φ)
x(0.7) = A * cos(ω(0.7) + 0)
We also know that x(0.7) = 2.0 m, as stated in the problem.
2.0 = A * cos(0.7ω)
Now, let's find ω using another equation:
ω = √(k/m)
Where:
- k is the spring constant (0.25 N/m) and
- m is the mass (0.3 kg).
ω = √(0.25 / 0.3)
ω = √0.833
ω ≈ 0.912
Now we can substitute ω in the equation:
2.0 = A * cos(0.7 * 0.912)
Using a calculator, solve for cos(0.637).
2.0 = A * cos(0.637)
Divide both sides by cos(0.637) to solve for A:
A = 2.0 / cos(0.637)
Using a calculator, find the value of A.
A ≈ 2.624 m
Therefore, the amplitude of oscillation is approximately 2.624 meters.