Time period for a Simple Harmonic Oscillator to go from 0-1m

[Ethan Godden]

Homework Statement

A particle with a mass(m) of 0.500kg is attached to a horizontal spring with a force constant(k) of 50.0N/m. At the moment t=0, the particle has its maximum speed of 20m/s and its moving to the left. Find the minimum time interval required for the particle to move from position x=0 to x=1.00m.

Homework Equations

General Equation of Motion: x(t)=(A)cos(ωt +Φ) where A is the amplitude or maximum position, ω is the angular frequency, and Φ is the phase constant
ω=√(k/m)
v_max=ωA

The Attempt at a Solution

ω=√(k/m)=√(50/0.5) = 10.0 rad/s
v_max=ωA → A=(v_max/ω)=20/10=2.00m

Since at t=0, the particle is at its maximum speed, the phase constant must make cosine=0 as the particles maximum speed is at position x=0, so the phase constant must be π/2.

Equation of Motion: 2cos(10t+π/2)=x(t)

I believe the solution from here it to make x(t)=1.00m as the particle is at x=0 at t=0, so the time at x(t)=1.00m is the time it takes to go from x=0 to x=1.00.

2cos(10t+π/2)=1.00 → cos^-1(1/2)=10t+π/2 → 10t= π/3 - π/2 → t= -π/60=-0.0524s.

I believe the answer is just the absolute value of this. When I input t=0.0524s, i get x(t)=-1.00. Shouldn't this be equivalent to the time it takes to get to +1.00m.

Apparently, the answer is 0.105s which is noticed is double the magnitude of my answer. I am pretty sure this is not a coincidence. It would be greatly appreciated if someone could tell me where I am going wrong.

Thank You,

 

[collinsmark]

Homework Statement

A particle with a mass(m) of 0.500kg is attached to a horizontal spring with a force constant(k) of 50.0N/m. At the moment t=0, the particle has its maximum speed of 20m/s and its moving to the left. Find the minimum time interval required for the particle to move from position x=0 to x=1.00m.

Homework Equations

General Equation of Motion: x(t)=(A)cos(ωt +Φ) where A is the amplitude or maximum position, ω is the angular frequency, and Φ is the phase constant
ω=√(k/m)
v_max=ωA

The Attempt at a Solution

ω=√(k/m)=√(50/0.5) = 10.0 rad/s
v_max=ωA → A=(v_max/ω)=20/10=2.00m

Since at t=0, the particle is at its maximum speed, the phase constant must make cosine=0 as the particles maximum speed is at position x=0, so the phase constant must be π/2.

Equation of Motion: 2cos(10t+π/2)=x(t)

I believe the solution from here it to make x(t)=1.00m as the particle is at x=0 at t=0, so the time at x(t)=1.00m is the time it takes to go from x=0 to x=1.00.

2cos(10t+π/2)=1.00 → cos^-1(1/2)=10t+π/2 → 10t= π/3 - π/2 → t= -π/60=-0.0524s.

The way I interpret the problem statement, you don't want to start the "timer" (so to speak) at the moment $t = 0$. That's because at time $t = 0$, the particle is moving to the left. The way I interpret the problem, it's okay to wait until the particle comes back to $x = 0$ such that it's moving to the right; only then begin the time interval. But I suppose your approach should work too, leveraging symmetry.

I believe the answer is just the absolute value of this. When I input t=0.0524s, i get x(t)=-1.00. Shouldn't this be equivalent to the time it takes to get to +1.00m.

I think your logic is valid due to symmetry. I might have instead tried to used different points in time such that I didn't have to take an absolute value, but that's just me.

Apparently, the answer is 0.105s which is noticed is double the magnitude of my answer.

All that said, I don't think the given answer is correct. I came up with the same answer that you did. Either we are right or we are misinterpreting the problem statement.

 

[TSny]

I also believe Ethan's answer is correct.

If you are familiar with the idea that SHM is just a projection of uniform circular motion, then you can see that when the particle in SHM moves from x = 0 to x = 1 m, the point on the reference circle moves through 30 degrees, or 1/12 of a circle. So, the time required is 1/12 of the period of the SHM.