Simple harmonic motion equation

[jumptheair]

Homework Statement

An object is undergoing simple harmonic motion. The graph shows the position of the object as a function of time. What is the first time after t = 0 s when the speed reaches a maximum?

http://www.learning.physics.dal.ca/dalphysicslib/Graphics/Gtype26/shm.47.jpeg

Homework Equations

I don't know what equation to use to figure this out. φ = φo + wt perhaps?

The Attempt at a Solution

Im told that position at t=0 gives phi_0. But still do not know how to use the equation nor figure out any of the variables. Help is needed!

 

[rl.bhat]

From the graph we get the following information. At t = 0 x = 1.5 cm, maximum displacent = 3cm, and period = 2s.
Wright the equation of simple harmonic motion x = Asin(wt + phi). In this put A = 3cm, t = 0 and x = 1.5 cm find phi. Next take derivative of the equation, which gives the velocity and put it equal to zero. And find t.

 

[ogb p]

I would first identify the type of motion as simple harmonic motion and then use the appropriate equation for this type of motion: x(t) = A*cos(ωt + φ). In this equation, A represents the amplitude of the motion, ω represents the angular frequency, t represents time, and φ represents the phase angle.

To determine the first time when the speed reaches a maximum, we can use the relationship between position and velocity in simple harmonic motion: v(t) = -A*ω*sin(ωt + φ). The maximum speed occurs when the sine function is equal to 1, which happens when ωt + φ = π/2. Therefore, the first time when the speed reaches a maximum is when ωt + φ = π/2, or when t = (π/2 - φ)/ω.

To find the value of φ, we can use the given information that at t = 0, the position is equal to φ_0. Therefore, we can substitute t = 0 into the equation for position to get φ_0 = A*cos(φ), which means φ = cos^-1(φ_0/A).

Finally, we can substitute this value of φ into the equation for t to get the first time when the speed reaches a maximum: t = (π/2 - cos^-1(φ_0/A))/ω.

In conclusion, the first time when the speed reaches a maximum can be determined by using the equation for simple harmonic motion and the given information about position at t = 0.