A 0.46 kg mass oscillates in simple harmonic motion on a spring that has a spring constant of 2080 N/m. The spring was initially compressed 9.4 cm (a negative displacement) at t = 0 s. What's the velocity in m/s of the mass at t = 0.24 s?
9.4 cm = 0.094 m
x = -0.094*cos(w*t)
v = dx/dt = 0.094*sin(w*t)
where w = (k/m)^0.5
x is the x position; v is the speed; w is the angular velocity; k is the spring constant; m is the mass
at t = 0.24 s,
v = 0.094*sin(((2080/0.46)^0.5)*0.24)
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To find the velocity of the mass at a particular time, we need to use the equations of simple harmonic motion. Specifically, we can use the equation for velocity in simple harmonic motion:
v = Aωsin(ωt + φ)
Where:
- v is the velocity of the mass
- A is the amplitude of the motion (maximum displacement from equilibrium)
- ω is the angular frequency of the motion
- t is the time
- φ is the phase constant (initial phase angle)
In this case, we are given the mass (m = 0.46 kg), the spring constant (k = 2080 N/m), and the initial displacement (x = -0.094 m). We can calculate the angular frequency (ω) using the formula:
ω = √(k/m)
Substituting the given values:
ω = √(2080 N/m / 0.46 kg)
= √4521.74
≈ 67.32 rad/s
We are also given t = 0.24 s. To find the initial phase angle (φ), we need to determine the phase constant at t = 0:
x = A sin φ
Substituting the given values:
-0.094 m = A sin φ
Since the spring was initially compressed, the displacement is negative. This means the amplitude (A) will also be negative. Assuming A = -0.094 m, we can rearrange the equation to find sin φ:
sin φ = x / A
= -0.094 m / -0.094 m
= 1
Since sin φ = 1, this implies that the phase constant φ is 90 degrees or π/2 radians.
Now, let's plug in the values into the equation for velocity:
v = Aωsin(ωt + φ)
= (-0.094 m) * (67.32 rad/s) * sin(67.32 rad/s * 0.24 s + π/2)
Using a calculator:
v ≈ (-0.094) * (67.32) * sin(16.15728 + π/2)
≈ (-0.094) * (67.32) * sin(16.15728 + 1.5708)
≈ (-0.094) * (67.32) * sin(17.72808)
≈ (-0.094) * (67.32) * 0.3031
≈ -1.9824 m/s
Therefore, the velocity of the mass at t = 0.24 s is approximately -1.9824 m/s.