Point P On the Unit Circle

[harpazo]

Use the picture below to answer parts A-D.

Suppose that the point P travels counterclockwise around the unit circle at a constant angular speed of pi/3 radians per second. Assume that at time t = 0 sec, the location of P is (1, 0).

A. Complete the table. I am confused concerning the location of t in the picture.

B. Draw a sketch showing the location points P and Q when t = 1 second. I need the set up for part B.

C. If you look back at the table in part A, you'll see that the relation between theta and t is theta = (pi•t)/3. Thus, the x-coordinate of Q at time t is given by x = cos (pi/3). This tells us that as the point P moves around the circle at a constant angular speed, the point Q oscillates in simple harmonic motion along the x-axis. Graph the function for two complete cycles, beginning at t = 0. Specify the amplitude, the period and the frequency for the motion.

D. Using calculus, it can be shown that the velocity of the point Q at time t is given by
v = (-pi/3)•sin(pi•t/3). Graph this function for two complete cycles, beginning at t = 0.

Note: I want to do the math work here. Looking for the set up and SET UP ONLY for parts A through D. I will do the rest. Thank you. Happy Sunday to all of you!

 

[topsquark]

Use the picture below to answer parts A-D.

Suppose that the point P travels counterclockwise around the unit circle at a constant angular speed of pi/3 radians per second. Assume that at time t = 0 sec, the location of P is (1, 0).

Given an angular speed [math]\omega = \pi / 3[/math] rad/s, then [math]\theta = \omega t = ( \pi t )/3[/math] rad.

This is directly from the definition of angular speed. The problem assumes you already know that.

Can you finish knowing this?

-Dan

 

[harpazo]

Given an angular speed [math]\omega = \pi / 3[/math] rad/s, then [math]\theta = \omega t = ( \pi t )/3[/math] rad.

This is directly from the definition of angular speed. The problem assumes you already know that.

Can you finish knowing this?

-Dan

I will work this out on paper and return here to continue if I get stuck somewhere along the way.

 

[HallsofIvy]

Use the picture below to answer parts A-D.

Suppose that the point P travels counterclockwise around the unit circle at a constant angular speed of pi/3 radians per second. Assume that at time t = 0 sec, the location of P is (1, 0).

A. Complete the table. I am confused concerning the location of t in the picture.

t is time! There is no "time" on the picture! There is no "t in the picture".
What is shown in the picture is angles. You are told that there is "constant angular speed of pi/3 radians per second. For time t, the angle is (pi/3)t. That is the closest you can come to the "location of t in the picture". When t= 0, P is at (1, 0) (because you are told that). When t= 1 P will have moved angle (pi/3)(1)= pi/3 radians (60 degrees) from (1, 0). Where is that point? When to is 2 P will have moved angle (pi/3)(2)= 2pi/3 radians (120 degrees) from (1, 0). Where is that point?

B. Draw a sketch showing the location points P and Q when t = 1 second. I need the set up for part B.

?? There is no mention of a point "Q" before!

C. If you look back at the table in part A, you'll see that the relation between theta and t is theta = (pi•t)/3. Thus, the x-coordinate of Q at time t is given by x = cos (pi/3). This tells us that as the point P moves around the circle at a constant angular speed, the point Q oscillates in simple harmonic motion along the x-axis. Graph the function for two complete cycles, beginning at t = 0. Specify the amplitude, the period and the frequency for the motion.

Perhaps this (and Q) is included in a part you didn't copy?

D. Using calculus, it can be shown that the velocity of the point Q at time t is given by
v = (-pi/3)•sin(pi•t/3). Graph this function for two complete cycles, beginning at t = 0.

Note: I want to do the math work here. Looking for the set up and SET UP ONLY for parts A through D. I will do the rest. Thank you. Happy Sunday to all of you!

The "set up" is the main part of the "math"! What you are saying is that you want to be told exactly what to do and then you will do it. "Learning math" means learning to decide what to do!

 

[MarkFL]

The "set up" is the main part of the "math"!

 

[harpazo]

If I could do the set up on my own, especially WORD PROBLEM SET UP, I would be working as a math tutor part-time to make extra money. Soroban understood this to be a HUGE PROBLEM for me back in the day.

 

[harpazo]

t is time! There is no "time" on the picture! There is no "t in the picture".
What is shown in the picture is angles. You are told that there is "constant angular speed of pi/3 radians per second. For time t, the angle is (pi/3)t. That is the closest you can come to the "location of t in the picture". When t= 0, P is at (1, 0) (because you are told that). When t= 1 P will have moved angle (pi/3)(1)= pi/3 radians (60 degrees) from (1, 0). Where is that point? When to is 2 P will have moved angle (pi/3)(2)= 2pi/3 radians (120 degrees) from (1, 0). Where is that point?


?? There is no mention of a point "Q" before!


Perhaps this (and Q) is included in a part you didn't copy?


The "set up" is the main part of the "math"! What you are saying is that you want to be told exactly what to do and then you will do it. "Learning math" means learning to decide what to do!

1. Thank you for breaking down part A. I now understand what is going on there.

2. In terms of point Q, I will need to check the textbook again. As you said, there may missing information here. Wait! Point Q is shown in the picture provided. Let me go back to the chapter.

3. Read my reply to MarkFL stated below.

If I could do the set up on my own, especially WORD PROBLEM SET UP, I would be working as a math tutor part-time to make extra money. Former FMH member, Soroban, understood this to be a HUGE PROBLEM for me back in the day.

4. Soroban stressed the importance of learning the SET UP. In terms of word problems, creating a desired equation(s) leading to the right answer is CRUCIAL for anyone teaching math at any level. Do you agree?

 

[HallsofIvy]

Yes, Thank you. "Q", though not mentioned in the text, is in the pictures. I, mistakenly, looked only at the text of the problem. Q is the foot of the perpendicular to the x-axis from P. The y-coordinate of Q is obviously 0. Since this is a unit circle, the hypotenuse of that right triangle is 1 and Q is the vertex on the "near side". The x coordinate is the cosine of the angle.

 

[HallsofIvy]

As for (4) I would say that learning to create a desired equation is CRUCIAL for any one learning math.

 

[harpazo]

As for (4) I would say that learning to create a desired equation is CRUCIAL for any one learning math.

I will do my best to finish this problem on paper, and if needed, come back here to show where I am stuck. Most likely, I will get stuck. More later....

 

[harpazo]

Given an angular speed [math]\omega = \pi / 3[/math] rad/s, then [math]\theta = \omega t = ( \pi t )/3[/math] rad.

This is directly from the definition of angular speed. The problem assumes you already know that.

Can you finish knowing this?

-Dan

Using theta = (pi•t)/3, I got the following values for theta (in radians):

t..........theta (in radians)

0..........(0)
1..........(pi/3)
2..........(2pi/3)
3..........(pi)
4..........(4pi/3)
5..........(5pi/3)
6..........(2pi)
7..........(7pi)i/3)

Correct?

Can you set up the remaining parts for me to calculate?