Uniform Circular Motion question

[15ongm]

1. The problem

The answer is A.

Homework Equations

x = Acoswt
y = Asinwt

The Attempt at a Solution

C is not correct because the x & y positions are in terms of sin & cos, so the acceleration can't be constant.
D is not correct because x & y are oscillating in value.
E is not correct because y is changing with time

Therefore it's either A or B, which makes sense since x & y are oscillating, they have to form some sort of circle.

I'm stuck between A & B. How do you know the particle is moving with constant speed? B/c the acceleration is definitely not 0. In fact, how do you know these 2 functions describe a circle?

 

[sk1105]

The easiest way to realize that it's a circle is to pick values for A and $\omega$ and just plot a few points. The more technical explanation is that cos(x) decreases at the same time and rate as sin(x) increases and then they switch (if you plot these two functions it's easy to see), so y will increase while x decreases at the same rate and then the opposite will happen. The effect of this is to trace a circle.

It seems like you are confusing centripetal acceleration with tangential acceleration. Non-zero centripetal acceleration is what makes the path curved rather than straight, but the change in speed along the path (tangential velocity) is governed by tangential acceleration. If you read up on tangential velocity (I'm sure it's in your textbook somewhere) you should be able to see why the particle described above moves at a constant speed.

 

[AlephNumbers]

The easiest way to realize that it's a circle is to pick values for A and $\omega$ and just plot a few points. The more technical explanation is that cos(x) decreases at the same time and rate as sin(x) increases and then they switch (if you plot these two functions it's easy to see), so y will increase while x decreases at the same rate and then the opposite will happen. The effect of this is to trace a circle.

It seems like you are confusing centripetal acceleration with tangential acceleration. Non-zero centripetal acceleration is what makes the path curved rather than straight, but the change in speed along the path (tangential velocity) is governed by tangential acceleration. If you read up on tangential velocity (I'm sure it's in your textbook somewhere) you should be able to see why the particle described above moves at a constant speed.

There is an analytical way to solve this problem. I would definitely recommend taking the analytical route whenever possible.

How do you know the particle is moving with constant speed?

You have functions to describe the vertical and horizontal positions of the particle. Start by trying to find functions to describe the vertical and horizontal velocities of the particles.

how do you know these 2 functions describe a circle?

It will become apparent if you follow my above approach.

 

[BvU]

Hello 15,

It isn't mentioned explicitly in the problem statement, but you may assume A and $\omega$ are constant in time.
As the others say, drawing a figure is the most logical thing to do.
If you want the analytical approach, ask what the distance of (x,y) to (0,0) is, and ask what the angle of the vector (x,y) with the x-axis is.