# Simple Harmonic Motion - Determine quadrant for phase angle.

1. Jan 17, 2016

### OmniNewton

1. The problem statement, all variables and given/known data
When completing this problem I am able to find a value for the phase angle but am unsure of how to find the quadrant for the phase angle therefore unable to get the correct phase angle.

2. Relevant equations
Provided in the question: x = xmcos(wt+ phi)

3. The attempt at a solution
After doing a series of calculations I determined sin(phi) = -0.8

at t =0

v(0) = -xmwsin(w(0) + phi)
v(0) = -vmsin(phi)
4.0cm/s = (-5.0 cm/s)sin(phi)
sin(phi) = -4/5 = -0.80

Therefore, one answer is -0.93 rad for the phase angle, phi. However other phase angles would also make sin(phi) = -0.8 such as -2.21. How do I determine which angle in radians is correct?

Last edited: Jan 17, 2016
2. Jan 17, 2016

### TSny

One way you might decide is to consider the sign of the acceleration at t = 0.

3. Jan 17, 2016

### OmniNewton

Yeah I notice that since when I did my calculations I used t =0 that the acceleration at that point was positive do to the slope of the velocity graph being positive. However how would I determine the angle range? I think it may be important to also know the velocity is positive at t = 0 perhaps.

4. Jan 17, 2016

### TSny

To make sure you are on the right track, would you mind showing how you got sin(φ) = -.80? You should not have needed to know anything about the acceleration to get this.

From the graph, is the acceleration positive or negative at t = 0. This should give you some more information about which quadrant φ is in.

5. Jan 17, 2016

### OmniNewton

Thank you for the response here are my calculations:

at t =0

v(0) = -xmwsin(w(0) + phi)
v(0) = -vmsin(phi)
4.0cm/s = (-5.0 cm/s)sin(phi)
sin(phi) = -4/5 = -0.80

EDIT: The acceleration is positive at t = 0 and so is the velocity. How do these quantities apply to the quadrant?

6. Jan 17, 2016

### TSny

Great!

How do you determine the acceleration from a velocity vs time graph?

7. Jan 17, 2016

### OmniNewton

acceleration was determined by the slope of the velocity vs time graph. At t = 0 I had observed a negative slope therefore acceleration is actually negative at t=0. Sorry I made an error. How does this negative acceleration relate to phase angle?

8. Jan 17, 2016

### TSny

What is the mathematical expression for the acceleration as a function of time? What does it reduce to at t = 0?

9. Jan 17, 2016

### OmniNewton

a(t) = -xmw^2 cos(wt + phi)
a(0) = -xm(w^2)cos(phi)
a(0) = -xw^2

10. Jan 17, 2016

### TSny

OK to here.
What happened to cos(phi)? [EDIT: Oh, I see. You dropped the subscript m on the position. OK]

11. Jan 17, 2016

### OmniNewton

Well I substituted the displacement function x(0)=xmcos(phi) into the velocity equation. I just called this equation x instead of x(0)

12. Jan 17, 2016

### TSny

OK. From a(0) = -xm(w^2)cos(phi) can you deduce anything about the sign of cos(phi)?

13. Jan 17, 2016

### OmniNewton

since xm and w^2 are always positive the cos(phi) is also positive. Therefore, according to the CAST rule since cos(phi) is positive the quadrant must be either 1 or 4 but sin(phi) is negative so therefore phi is in quadrant for so phi= -0.92 radians (to 2 significant figures) is the answer?

14. Jan 17, 2016

### TSny

15. Jan 17, 2016

### OmniNewton

Actually did I make an error, should cos(phi) actually be negative do to the negative in front so the answer should be quadrant 3? since sin and cos are both negative in quadrant 3?

16. Jan 17, 2016

### TSny

Sorry, I'm not following how you are getting that cos(phi) should be negative. What is the sign of a(0)?

17. Jan 17, 2016

### OmniNewton

Not a problem let me try to explain.

xm is a positive quantity. Angular frequency, w , a positive quantity
a(0) = -xm(w^2)cos(phi)
a(0) = -(positive)(positive)^2(cos(phi))
negative * positive * positive = negative
that is,
a(0) = -cos(phi)

18. Jan 17, 2016

### TSny

Exactly.
So, what is the sign of cos(phi)?

19. Jan 17, 2016

### OmniNewton

Is it not negative? Or am I perhaps mislead by the negative sign in front of cos(phi)?

20. Jan 17, 2016

### TSny

If cos(phi) were negative, what would be the sign of -cos(phi)?