Uniform Circular Motion, Acceleration problem

In summary, at time t2 = 5.00 s, the cat’s velocity is V2 = (3.00 m/s)i + (4.00 m/s)j. The magnitude of the centripetal acceleration is a = V2/r, and the cat’s average acceleration is aavg = (5.00 m/s)2/4,8 m = 5,2 m/s2.
  • #1
No1_129848
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Homework Statement


A cat rides a merry-go-round turning with uniform circular motion. At time t1 = 2.00 s, the cat’s velocity is V1 = (3.00 m/s)i + (4.00 m/s)j , measured on a horizontal xy coordinate system. At t2 = 5.00 s, the cat’s velocity is V2 = (3.00 m/s)i + (4.00 m/s)j.
What are (a) the magnitude of the cat’s centripetal acceleration and (b) the cat’s average acceleration during the time interval t2 - t1, which is less than one period?

Homework Equations


[/B]
T = 2πr/V
a = V2/r

The Attempt at a Solution


[/B]
So, the first thing I did was sketch the situation, and the vectors seem to be on the opposite end of the circumference, so i tested for that:

V1 * V2 = V1x * V2x + V1y * V2y = 3*(-3) + 4*(-4) = -25
V1 * V2 = V1 * V2 CosΘ = √32+42*√(-3)2+(-4)2 = 25 CosΘ → CosΘ = -1 → Θ = Cos-1(-1) = 180°

So it's proven that the cat is in two different positions that cover half of the circumference, which means that
Δt = ½T
Δt = t2-t1 = 5s-2s = 3s
T = 2Δt = 3s * 2 = 6s
T = 2πr/V → r = TV/2π = 6s * 5 m/s / 2π = 4,8 m
a = V2/r → (5 m/s)2/4,8 m = 5,2 m/s2

So this should answer the first part of the problem, now I'm asked to find the average acceleration, and I'm not sure what approach is the correct one, I know the equation for the average acceleration is:

aavg = V2 - V1 / t2 - t1

And I am not sure if I should consider just the magnitude of the two velocities, which means that aavg = 0, or if I should solve in unit vector notation, doing the following:

aavg = V2 - V1 / t2 - t1 = (-3-3)i + (-4-4)j / 3s = -2i -2,67j → aavg = √(-2)2+(-2,67)2 = 3,3 m/s2

What is the correct way to answer the second question?

(also, kinda off topic, is there a way to write fractions and vectors on the forum?)

Thanks in advance, Ivan.
 
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  • #2
Umh, the spam filter is not letting me edit the post, but of course the equation for the average acceleration is

aavg = V2 - V1 / t2 - t1

And I have used that in the calculations.
 
  • #3
No1_129848 said:
At time t1 = 2.00 s, the cat’s velocity is V1 = (3.00 m/s)i + (4.00 m/s)j , measured on a horizontal xy coordinate system. At t2 = 5.00 s, the cat’s velocity is V2 = (3.00 m/s)i + (4.00 m/s)j.
Is this correct? It looks like V1 = V2 which means that the cat has made a complete revolution. Did you miss a negative sign somewhere?
 
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  • #4
Oh, indeed, both signs are negative in V2, i can't edit the OP, but the correct data is
V1 = (3.00 m/s)i + (4.00 m/s)j
V2 = (-3.00 m/s)i + (-4.00 m/s)j
 
  • #5
Part (a) looks fine.
You must use unit vector notation to find the angular acceleration. I would also say that, since the question asks for the average acceleration and not its magnitude, you should leave in unit vector form.
Yes, there is a way to write fractions and all sorts of other algebraic expressions. Click on the link "LaTeX", near bottom left next to the question mark.
 
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  • #6
Thanks for the help, much appreciated!
 

1. What is uniform circular motion?

Uniform circular motion is the motion of an object along a circular path at a constant speed.

2. How is acceleration calculated in uniform circular motion?

In uniform circular motion, the acceleration is constantly changing in direction but has a constant magnitude. It can be calculated using the formula a = v^2/r, where v is the speed of the object and r is the radius of the circular path.

3. How does centripetal force relate to uniform circular motion?

Centripetal force is the force that keeps an object moving in a circular path. In uniform circular motion, the centripetal force is always directed towards the center of the circle and is equal to the mass of the object multiplied by its centripetal acceleration.

4. What is the difference between tangential and radial acceleration in uniform circular motion?

Tangential acceleration is the component of acceleration that is tangent to the circular path and is responsible for changing the speed of the object. Radial acceleration is the component of acceleration that is directed towards the center of the circle and is responsible for changing the direction of the object's velocity.

5. How does the radius of the circular path affect the acceleration in uniform circular motion?

The acceleration in uniform circular motion is inversely proportional to the radius of the circular path. This means that as the radius increases, the acceleration decreases, and vice versa. This is because a larger radius means a larger distance to cover in the same amount of time, resulting in a lower speed and therefore a lower acceleration.

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