Relationship between linear velocity and angular velocity

In summary, linear velocity and angular velocity are related concepts in rotational motion. Linear velocity (v) refers to the speed of an object moving along a path, while angular velocity (ω) measures the rate of rotation around an axis. The relationship between the two is defined by the formula v = rω, where r represents the radius or distance from the axis of rotation to the point of interest. This equation illustrates that an increase in angular velocity results in a proportional increase in linear velocity, depending on the radius.
  • #1
good jelly
2
0
Homework Statement
Find the relationship between the linear velocity and angular velocity for mass B which is moving about mass A (mass A is stationary).
Relevant Equations
v = wr
I find the velocity vector that is perpendicular to the radius which is v*cos(theta). Once I obtained the velocity that is perpendicular to the radius I used the equation : v = wr. But my answer seems to be wrong, the actual answer to this question is v = wr*cos(theta). Why?

imagephysicsproblemletsgo.jpg
 
Physics news on Phys.org
  • #2
good jelly said:
the actual answer to this question is v = wr*cos(theta)
No it isn’t and it is easy to see that it isn’t by considering the limit ##\theta \to 0##.
 
  • #3
good jelly said:
Why?
Only the velocity component perpendicular to the radius (i.e. tangential to the path) counts. The radial component (towards or away from the center) does not count because it does not change the angle.
 
  • #4
@good jelly :
Welcome to PF!

Have you given the complete statement of the problem "word-for-word"?
Based on your diagram, your equation ##v_B \cos \theta = \omega R## looks correct.
 
  • #5
kuruman said:
Only the velocity component perpendicular to the radius (i.e. tangential to the path) counts. The radial component (towards or away from the center) does not count because it does not change the angle.
That wasn’t the question. That was exactly what OP did. The question was why the wrong answer provided by [unknown, probably teacher] is correct. It isn’t.
 
  • #6
Orodruin said:
That wasn’t the question. That was exactly what OP did. The question was why the wrong answer provided by [unknown, probably teacher] is correct. It isn’t.
You are right, that was not the question. One can only provide reassurance that OP's answer is correct without attempting to divine like a haruspex why the other answer is incorrect.
 
  • Haha
  • Like
Likes SammyS and haruspex
  • #7
Orodruin said:
No it isn’t and it is easy to see that it isn’t by considering the limit ##\theta \to 0##.
Isn't that limit the one case in which the two formulae both yield ##v=\omega r##?
Maybe you meant ##\theta\rightarrow \pi/2##.
 
  • #8
haruspex said:
Isn't that limit the one case in which the two formulae both yield ##v=\omega r##?
Maybe you meant ##\theta\rightarrow \pi/2##.
Yes, definitely. Good catch.
 

FAQ: Relationship between linear velocity and angular velocity

What is the relationship between linear velocity and angular velocity?

The relationship between linear velocity (v) and angular velocity (ω) can be expressed by the equation v = r * ω, where r is the radius of the circular path. This means that linear velocity is directly proportional to both the radius and the angular velocity of an object moving in a circular path.

How do you calculate linear velocity from angular velocity?

To calculate linear velocity from angular velocity, you can use the formula v = r * ω. Here, you need to know the radius (r) of the circular path and the angular velocity (ω) in radians per second. Multiply these two values to find the linear velocity.

What units are used for linear velocity and angular velocity?

Linear velocity is typically measured in meters per second (m/s) or kilometers per hour (km/h), while angular velocity is measured in radians per second (rad/s) or degrees per second (°/s). It's important to ensure that the units are compatible when using the relationship between them.

Can linear velocity be greater than angular velocity?

Yes, linear velocity can be greater than angular velocity, but this comparison is not straightforward since they measure different quantities. Linear velocity depends on both the angular velocity and the radius of the circular path. For a larger radius, even a relatively small angular velocity can result in a high linear velocity.

How does the radius affect the relationship between linear and angular velocity?

The radius plays a crucial role in the relationship between linear and angular velocity. A larger radius results in a greater linear velocity for a given angular velocity, as indicated by the formula v = r * ω. Conversely, for a fixed linear velocity, increasing the radius will decrease the angular velocity, since ω = v / r.

Similar threads

Circular motion doubt: Angular velocity vector for general planar motion about a point in the plane
Replies
6
Views
463
System of two wheels of different sizes with an axle through their centers
2
Replies
67
Views
3K
Relation between the velocity vector and the acceleration vector of an object
Replies
1
Views
343
What mistake did I make in finding the net force on a rotating disc?
Replies
1
Views
1K
Volumetric flow of a overflowing cone
Replies
6
Views
1K
Circular motion of a Weightless rod
Replies
18
Views
558
Find the acceleration in circular motion
Replies
6
Views
1K
Circular motion, coin on a rotating disk
Replies
2
Views
399
Finding max velocity for a kart on a circular, banked track
Replies
2
Views
1K
How Do You Correctly Cancel Angular Momentum in Physics Problems?
Replies
3
Views
858

Hot Threads

Recent Insights

Back
Top