Uniform Circular Motion, Acceleration problem

[No1_129848]

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

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T = 2πr/V
a = V²/r

The Attempt at a Solution

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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 = V_1x * V_2x + V_1y * V_2y = 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:

a_avg = V₂ - V₁ / t₂ - t₁

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

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

 

[No1_129848]

Umh, the spam filter is not letting me edit the post, but of course the equation for the average acceleration is

a_avg = V₂ - V₁ / t₂ - t₁

And I have used that in the calculations.

 

[kuruman]

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?

 

[No1_129848]

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

 

[kuruman]

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.

 

[No1_129848]

Thanks for the help, much appreciated!