Part (a) of the figure below is a partial graph of the position function x(t) for a simple harmonic oscillator with an angular frequency of 1.00 rad/s; Part (b) of the figure is a partial graph of the corresponding velocity function v(t). The vertical axis scales are set by xs = 7.5 cm and vs = 2.5 cm/s. What is the phase constant of the SHM if the position function x(t) is given by the form x = xmcos(ωt + ϕ)?

Question

Part (a) of the figure below is a partial graph of the position function x(t) for a simple harmonic oscillator with an angular frequency of 1.00 rad/s; Part (b) of the figure is a partial graph of the corresponding velocity function v(t). The vertical axis scales are set by xs = 7.5 cm and vs = 2.5 cm/s. What is the phase constant of the SHM if the position function x(t) is given by the form x = xmcos(ωt + ϕ)?

units in radians

Answer 1

There is no "figure below". The phase constant for the position function you have listed is ϕ

Please show your own work in any future posts.

Answer 2

To find the phase constant of SHM, we need to analyze the position function and the given graph.

The position function can be written as x = xmcos(ωt + ϕ), where x is the displacement from the equilibrium position, xm is the amplitude of oscillation, ω is the angular frequency, t is the time, and ϕ is the phase constant.

In the given graph, Part (a), we can see that the displacement is maximum when t = 0. This means that at t = 0, the value of cos(ωt + ϕ) should be equal to 1, because cos(0) = 1.

From the graph, we can see that the displacement at t = 0 is xs = 7.5 cm. So we substitute these values into the position function:

7.5 cm = xmcos(0 + ϕ)
7.5 cm = xmcos(ϕ)

Since cos(0) = 1, we can rewrite the equation as:

7.5 cm = xm * 1
xm = 7.5 cm

Now we have the value of xm, which is the amplitude of oscillation.

To find the phase constant ϕ, we can use the given graph, Part (b), which shows the velocity function v(t).

The velocity function is the derivative of the position function, v(t) = dx/dt. Thus, the maximum value of the velocity occurs when the position function is passing through equilibrium, which is at t = 0.

From the graph, we can see that the maximum velocity occurs at t = 0, and its value is vs = 2.5 cm/s. Using the position function, we can differentiate it to find the velocity function:

v(t) = dx/dt = -xmωsin(ωt + ϕ)

At t = 0, we have:

vs = -xmωsin(0 + ϕ)
vs = -xmωsin(ϕ)

Solving this equation, we can find the value of ϕ.

ϕ = arcsin(-vs / (xmω))

Substituting the given values, ϕ = arcsin(-2.5 cm/s / (7.5 cm * 1.00 rad/s)).

Using a calculator, we can find the value of ϕ. Note that the result will be in radians since the units for ω are radians.

Once you calculate the value of ϕ, you will have determined the phase constant of the SHM.