Relative Velocity of Particles a & b in Angular Motion

In summary, the particles ##a## and ##b## move in opposite directions around a circle with angular speed ##\omega## and are initially at the point ##\vec{r} = l \hat{\jmath}##. The velocity of ##a## relative to ##b##, as a function of time, is ##\vec{v}_{a/b} = 2l\omega \cos{\omega t} \hat{\imath}##.
  • #1
PFuser1232
479
20

Homework Statement



Particles ##a## and ##b## move in opposite directions around a circle with angular speed ##ω##. At ##t = 0## they are both at the point ##\vec{r} = l \hat{\jmath}## where ##l## the radius of the circle.

Find the velocity of ##a## relative to ##b##

Homework Equations



$$\hat{\theta} = -\hat{\imath} \sin{\theta} + \hat{\jmath} \cos{\theta}$$
$$\vec{v}_{a/b} = \vec{v}_{a/O} - \vec{v}_{b/O}$$
$$\theta = \theta_0 + \dot{\theta}t$$
$$\vec{v} = \dot{\theta} r \hat{\theta}$$

The Attempt at a Solution



First, I found the tangential unit vector as a function of time for each of the particles, setting the initial angle to ##\frac{\pi}{2}## for both ##a## and ##b##. Also, according to the diagram shown in the source, ##a## is moving clockwise (towards the ##\hat{\imath}## unit vector) and ##b## is moving anticlockwise; so I set ##\dot{\theta}## equal to ##\omega## for ##b## and ##-\omega## for ##a##.
I then found the tangential velocities of both ##a## and ##b##, setting ##r## equal to ##l## for both. After doing the math and cleaning up the vector components using trigonometric identities, I got:

$$\vec{v}_{a/b} = 2l\omega \cos{\omega t} \hat{\imath}$$

I have no answer booklet though, so I don't know whether that's correct. Did I get it right, or did I go wrong somewhere?
 
Last edited:
Physics news on Phys.org
  • #3
Are you supposed to find their relative velocities as a function of time, or are you just supposed to find it at t = 0?

Chet
 
  • #4
Chestermiller said:
Are you supposed to find their relative velocities as a function of time, or are you just supposed to find it at t = 0?

Chet
Chestermiller said:
Are you supposed to find their relative velocities as a function of time, or are you just supposed to find it at t = 0?

Chet
Chestermiller said:
Are you supposed to find their relative velocities as a function of time, or are you just supposed to find it at t = 0?

Chet

I wasn't sure, so I figured writing the velocity of ##a## relative to ##b## as a function of time would be better, since I could just set ##t## equal to zero and get the initial relative velocity (##2l\omega \hat{\imath}##).
 
  • #5
Nice job.

Chet
 

1. What is the definition of relative velocity in angular motion?

The relative velocity of particles a and b in angular motion is the rate at which their positions change with respect to each other as they move along a circular path. It is the difference in the angular velocities of the two particles.

2. How is relative velocity of particles a and b calculated in angular motion?

The relative velocity of particles a and b can be calculated by taking the difference between their angular velocities and multiplying it by the radius of the circular path. This can be represented by the formula Vab = (ωb - ωa) * r, where Vab is the relative velocity, ωb is the angular velocity of particle b, ωa is the angular velocity of particle a, and r is the radius of the circular path.

3. What is the significance of relative velocity in angular motion?

Relative velocity in angular motion is important because it helps us understand the relative motion of particles on a circular path. It allows us to determine the speed and direction of each particle with respect to the other, which is crucial in analyzing and predicting the behavior of systems in circular motion.

4. Can the relative velocity of particles a and b in angular motion ever be zero?

Yes, the relative velocity of particles a and b can be zero in certain cases. This happens when the two particles are moving at the same angular velocity along the circular path, meaning their positions are not changing relative to each other. In this case, the relative velocity would be zero, and the particles would be moving together in perfect synchronization.

5. How does the direction of the relative velocity of particles a and b in angular motion affect their motion?

The direction of the relative velocity of particles a and b in angular motion determines whether the particles are moving closer together or farther apart. If the relative velocity is positive, it means that the particles are moving closer together, and if it is negative, they are moving farther apart. This affects the overall shape and behavior of the circular path and can also impact the forces acting on the particles.

Similar threads

Replies
20
Views
788
  • Introductory Physics Homework Help
Replies
4
Views
966
  • Introductory Physics Homework Help
Replies
1
Views
780
  • Introductory Physics Homework Help
Replies
2
Views
744
  • Introductory Physics Homework Help
Replies
9
Views
608
  • Introductory Physics Homework Help
Replies
17
Views
278
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
14
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
392
  • Introductory Physics Homework Help
Replies
5
Views
517
Back
Top