Kinematics of Rigid Body Plane Motion

[Auburn2017]

Homework Statement

The wheel shown rotates about point O.
Point A has a velocity=-7j in/sec.
Point B has a tangential velocity=-4i in/sec^2
Determine the absolute acceleration of Point C located on the circle.
radius=6 in
You will have to look at the figure for more clarification please.

Homework Equations

r_A=r_B+r_A/B
v_A=v_B+v_A/B
a_A=a_B+a_A/B
v_A/B=ω×r_A/B
a_A/B=(a_A/B)_n+(a_A/B)_t
(a_A/B)_n=ω×(ω×r_A/B)
(a_A/B)_t=α×r_A/B

The Attempt at a Solution

I tried finding the velocity of C relative to A. I feel like there is an easy way to find the angular acceleration and angular velocity of the wheel but I don't really know how. Thanks for your help.

 

[TSny]

The wheel is considered to be a rigid body rotating about a fixed axis. How does the absolute speed of point C compare to the absolute speed of A? How does the magnitude of the absolute tangential acceleration of C compare to that of B?

 

[Auburn2017]

The wheel is considered to be a rigid body rotating about a fixed axis. How does the absolute speed of point C compare to the absolute speed of A? How does the magnitude of the absolute tangential acceleration of C compare to that of B?

Since all three points are on the edge of the wheel then they should all have the same absolute acceleration and velocity.

 

[TSny]

The magnitudes of the velocity and tangential acceleration will be the same. Directions?

 

[Auburn2017]

The magnitudes of the velocity and tangential acceleration will be the same. Directions?

velocity CW
acceleration CCW

 

[TSny]

OK. But more specifically, can you express the tangential acceleration vector of point C in terms of the unit vectors $\hat{i}$ and $\hat{j}$?

 

[Auburn2017]

No I can't. Isn't that what the problem is asking?

 

[TSny]

No, there's more to the problem than just doing that. Does point C have any other component of acceleration besides tangential?

 

[Auburn2017]

No, there's more to the problem than just doing that. Does point C have any other component of acceleration besides tangential?

Yes it has a normal acceleration toward the center of the circle. I guess the tangential acceleration would be a_tcos60° i and a_tsin60° j

 

[TSny]

You need to draw the tangential and normal acceleration vectors for point C on your diagram. The tangential acceleration of C is not vertically upward (i.e., not +j) and it's not "to the right" (i.e., not i).

Don't worry about the "hats". You could use bold type instead.

 

[Auburn2017]

You need to draw the tangential and normal acceleration vectors for point C on your diagram. The tangential acceleration of C is not vertically upward (i.e., not +j).

Don't worry about the "hats". You could use bold type instead.

look at my edited answer please

 

[TSny]

Yes it has a normal acceleration toward the center of the circle. I guess the tangential acceleration would be a_tcos60° i and a_tsin60° j

That's getting close. But you need to check if 60^o is correct here.

 

[Auburn2017]

That's getting close. But you need to check if 60^o is correct here.

yes it is correct

 

[TSny]

Did you actually draw the tangential acceleration vector at C on your diagram? What angle does it make to the horizontal?

 

[Auburn2017]

Did you actually draw the tangential acceleration vector at C on your diagram? What angle does it make to the horizontal?

Yes I drew it. I'm not sure how to tell the angle though...

 

[TSny]

You'll need to use a little geometry.

 

[Auburn2017]

You'll need to use a little geometry.

I'm going with 30 degrees

 

[TSny]

Did you work it out, or is that a guess?

Look at the figure below. What is the angle $\phi$? Form $\phi$ do you see how to get $\theta$?

 

[Auburn2017]

Did you work it out, or is that a guess?

Look at the figure below. What is the angle $\phi$? Form $\phi$ do you see how to get $\theta$?
View attachment 103452

Yes theta would equal 30 degrees

 

[TSny]

Good. Repeat for the normal acceleration.

 

[Auburn2017]

normal acceleration would be a_ncos60°-i and a_nsin60°+j

 

[TSny]

Good.

 

[Auburn2017]

Good.

Am I correct in saying that I will need to know the velocity of point C to find the final answer?

 

[TSny]

The speed at C should be sufficient. Don't worry about the velocity (vector) at C.

 

[Auburn2017]

The speed at C should be sufficient. Don't worry about the velocity (vector) at C.

And I will get the speed by using the relative velocities of the points B and C

 

[TSny]

How does the speed at C compare to the speed at A?

 

[Auburn2017]

How does the speed at C compare to the speed at A?

it is the speed of A plus speed of C relative to A

 

[TSny]

You need the absolute speed of C. How does the absolute speed of C compare to the absolute speed of A? See posts #3 and #4.

 

[Auburn2017]

You need the absolute speed of C. How does the absolute speed of C compare to the absolute speed of A? See posts #3 and #4.

it is the same as A but in a different direction

 

[TSny]

Speed doesn't have a direction.