Uniform circular motion of a particle

[southernbelle]

Homework Statement

A particle is moving at a constant speed around a circle of radius 1.5 meters, and it completes 4 revolutions in 6 seconds.
a) What is its speed?
b) What is its radial acceleration?
c) In which direction does ar point?
d) What is it tangential acceleration, at?

Homework Equations

ac=v²/r
T=2pie times r/v
at= derivative of v/t
ar = -v²/r
ac = ar + at

The Attempt at a Solution

a.
6=8(pie)(1.5)/v
answer: 2pie
v=6.28 m/s

b.
ar = -v²/r
ar = 39.43/1.5
ar = 26.29 m/s²

c. I am not sure how to get the direction.

d. ac= ar + at
26.29 = 26.29 + at
at = 0?

 

[S[e^x]=f(u)^n]

think about what would happen if it accelerated constantly tangentially, and describe that motion, and see if it fits with your problem

 

[baba89]

I would like to start by clarifying the meaning of the terms used in this problem. Uniform circular motion refers to the movement of an object at a constant speed along a circular path. In this case, the particle is moving at a constant speed of 6.28 m/s along a circle with a radius of 1.5 meters. The term "radial acceleration" refers to the acceleration towards or away from the center of the circle, while "tangential acceleration" refers to the acceleration along the tangent of the circle.

a. To find the speed of the particle, we can use the equation T=2πr/v, where T is the time it takes to complete one revolution. In this case, T=6/4=1.5 seconds. Plugging in the values, we get v=2π(1.5)/1.5=2π m/s=6.28 m/s.

b. The radial acceleration can be calculated using the equation ar=v^2/r. Plugging in the values, we get ar=(6.28)^2/1.5=26.29 m/s^2. This means that the particle is accelerating towards the center of the circle at a rate of 26.29 m/s^2.

c. The direction of ar can be determined by the negative sign in front of the equation. This indicates that the acceleration is directed towards the center of the circle, or in other words, in the opposite direction of the particle's velocity.

d. To find the tangential acceleration, we can use the equation at=dv/dt, where dv is the change in velocity and dt is the change in time. In this case, the particle is moving at a constant speed, so there is no change in velocity and therefore no tangential acceleration (at=0). This makes sense since the particle is moving along the tangent of the circle at a constant speed, so there is no change in its tangential direction.

In summary, the particle is moving at a speed of 6.28 m/s, with a radial acceleration of 26.29 m/s^2 towards the center of the circle, and no tangential acceleration.