Kinematics of Rigid Body Plane Motion

AI Thread Summary
The discussion revolves around determining the absolute acceleration of Point C on a rotating wheel, given the velocities of Points A and B. Key equations for rigid body motion are referenced, including those for relative velocity and acceleration. Participants clarify the relationship between the speeds and accelerations of Points A, B, and C, emphasizing that while their speeds are equal, their velocities differ in direction. The importance of distinguishing between speed (scalar) and velocity (vector) is highlighted, along with the need to calculate normal and tangential accelerations. The conversation concludes with the user successfully finding the necessary values for the components of acceleration.
Auburn2017
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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


rA=rB+rA/B
vA=vB+vA/B
aA=aB+aA/B
vA/B=ω×rA/B
aA/B=(aA/B)n+(aA/B)t
(aA/B)n=ω×(ω×rA/B)
(aA/B)t=α×rA/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.
 

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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?
 
Last edited:
TSny said:
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.
 
The magnitudes of the velocity and tangential acceleration will be the same. Directions?
 
TSny said:
The magnitudes of the velocity and tangential acceleration will be the same. Directions?
velocity CW
acceleration CCW
 
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}##?
 
No I can't. Isn't that what the problem is asking?
 
No, there's more to the problem than just doing that. Does point C have any other component of acceleration besides tangential?
 
TSny said:
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 atcos60° i and atsin60° j
 
  • #10
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.
 
  • #11
TSny said:
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
 
  • #12
Auburn2017 said:
Yes it has a normal acceleration toward the center of the circle. I guess the tangential acceleration would be atcos60° i and atsin60° j
That's getting close. But you need to check if 60o is correct here.
 
  • #13
TSny said:
That's getting close. But you need to check if 60o is correct here.
yes it is correct
 
  • #14
Did you actually draw the tangential acceleration vector at C on your diagram? What angle does it make to the horizontal?
 
  • #15
TSny said:
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...
 
  • #16
You'll need to use a little geometry.
 
  • #17
TSny said:
You'll need to use a little geometry.
I'm going with 30 degrees
 
  • #18
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##?
upload_2016-7-18_20-21-15.png
 
  • #19
TSny said:
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
 
  • #20
Good. Repeat for the normal acceleration.
 
  • #21
normal acceleration would be ancos60°-i and ansin60°+j
 
  • #22
Good.
 
  • #23
TSny said:
Good.
Am I correct in saying that I will need to know the velocity of point C to find the final answer?
 
  • #24
The speed at C should be sufficient. Don't worry about the velocity (vector) at C.
 
  • #25
TSny said:
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
 
  • #26
How does the speed at C compare to the speed at A?
 
  • #27
TSny said:
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
 
  • #28
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.
 
  • #29
TSny said:
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
 
  • #30
Speed doesn't have a direction.
 
  • #31
TSny said:
Speed doesn't have a direction.
This is just not making sense to me.
 
  • #32
Speed is the magnitude of the velocity. Velocity is a vector. Speed is a scalar.

So, the velocity of C is not the same as the velocity of A. But the speed of C does equal the speed of A.
 
  • #33
TSny said:
Speed is the magnitude of the velocity. Velocity is a vector. Speed is a scalar.

So, the velocity of C is not the same as the velocity of A. But the speed of C does equal the speed of A.
but you can't get acceleration from speed
 
  • #34
You can get the magnitude of the normal acceleration from the speed and the radius. Normal acceleration is also called centripetal acceleration.
 
  • #35
TSny said:
You can get the magnitude of the normal acceleration from the speed and the radius. Normal acceleration is also called centripetal acceleration.
how would you find the angular acceleration and velocity of the wheel?
 
  • #36
Are you familiar with the formula acentripetal = v2/r ?

Or are you required to solve this problem using only the equations that you listed:

rA=rB+rA/B
vA=vB+vA/B
aA=aB+aA/B
vA/B=ω×rA/B
aA/B=(aA/B)n+(aA/B)t
(aA/B)n=ω×(ω×rA/B)
(aA/B)t=α×rA/B
 
  • #37
TSny said:
Are you familiar with the formula acentripetal = v2/r ?

Or are you required to solve this problem using only the equations that you listed:
Yes I am familiar with such equation
 
  • #38
Great! What do you get for the magnitude of the centripetal acceleration?
 
  • #39
TSny said:
Great! What do you get for the magnitude of the centripetal acceleration?
49/6
 
  • #40
√(49)/6
Why the square root?
 
Last edited:
  • #41
TSny said:
Why the square root?
brain fart just 49/6
 
  • #42
OK. All you need to do is put it all together now.
 
  • #43
TSny said:
OK. All you need to do is put it all together now.
the magnitude of the normal acceleration would be radius times angular acceleration which we know from point B correct?
 
  • #44
Yes, but why go there? You already have values for an and at. Use these in your earlier expressions to obtain the normal and tangential acceleration vectors in terms of i and j. For example, see your post #21.
 
  • #45
TSny said:
Yes, but why go there? You already have values for an and at. Use these in your earlier expressions to obtain the normal and tangential acceleration vectors in terms of i and j. For example, see your post #21.
But what is the value of an
 
  • #46
Post # 41 ??
 
  • #47
no no sorry at
 
  • #48
How does the magnitude of the tangential acceleration of C compare to the magnitude of the tangential acceleration of B?
 
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  • #49
TSny said:
How does the magnitude of the tangential acceleration of C compare to the magnitude of the tangential acceleration of B?
SAME
 
  • #50
Yes.
 
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