Uniform circular motion of a particle

In summary, the given position function describes a particle moving in a circular motion with radius R and angular velocity ω. At t=0, the particle starts at the positive x axis. To find when the particle first crosses the negative x axis, we can set the y-coordinate to 0 and solve for t, giving us a value of t=π/ω. For part b, the speed of the particle can be found by taking the magnitude of the velocity vector, which is given by |v(t)| = Rω. Finally, the magnitude of the acceleration of the particle can be found by taking the magnitude of the acceleration vector, which is given by |a(t)| = Rω².
  • #1
mclame22
13
0

Homework Statement



A particle's position is given by the formula
r(t) = Rcos(ωt)î + Rsin(ωt)ĵ

The particle's motion at t=0 can be described as a circle starting at time t=0 on the positive x axis.

a) When does the particle first cross the negative x axis?
b) Find the speed of the particle at time t.
(Express your answers in terms of some or all of the variables ω, R, and π.)

c) Find the magnitude of the acceleration of the particle as a function of time.
(Express your answer in terms of some or all of the variables R, ω, and t.)

Homework Equations



r(t) = Rcos(ωt)î + Rsin(ωt)ĵ
v(t) = -Rωsin(ωt)î + Rωcos(ωt)ĵ
a(t) = -Rω²[cos(ωt)î + sin(ωt)ĵ] = -ω²r(t)

The Attempt at a Solution



I'm really unsure about what to do for part a. I know it has something to do with the particle moving pi radians along the arc of the circle. Would π be the distance it travels? Would it then be
π = Rcos(ωt)î + Rsin(ωt)ĵ ?
But how would I solve this for t?

For part b, I found the velocity of the particle to be the v(t) formula above by taking the derivative of the given position function. Would I take the absolute value of the velocity function to find the speed? How would I go about doing that?

Finally, for part c, I found that the acceleration of the particle is the a(t) function above after taking the derivative of the velocity function, but once again I don't know how to get the "magnitude" of the acceleration. Would I take the absolute value?
 
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  • #2
mclame22 said:
I'm really unsure about what to do for part a. I know it has something to do with the particle moving pi radians along the arc of the circle. Would π be the distance it travels? Would it then be
π = Rcos(ωt)î + Rsin(ωt)ĵ ?
But how would I solve this for t?
That should be an 'r' not an 'n'. Notice that this equation is given in terms of the x and y (i and j) coordinates. What constraint can you make on the y-value for part a?

mclame22 said:
For part b, I found the velocity of the particle to be the v(t) formula above by taking the derivative of the given position function. Would I take the absolute value of the velocity function to find the speed? How would I go about doing that?
Again, this equation is parametric---describing a vector. How can you find the magnitude of that vector?---"absolute value" is insufficient, although it is often written as magnitude of v(t) = |v(t)|
 

Related to Uniform circular motion of a particle

1. What is uniform circular motion of a particle?

Uniform circular motion of a particle is the movement of a particle in a circular path at a constant speed. This means that the particle covers equal distances in equal amounts of time and its speed remains constant throughout the motion.

2. What is the difference between uniform circular motion and non-uniform circular motion?

The main difference between uniform and non-uniform circular motion is that in uniform circular motion, the speed of the particle remains constant, while in non-uniform circular motion, the speed changes at different points in the motion. This can be due to changes in the direction or magnitude of the velocity.

3. How is centripetal force related to uniform circular motion?

Centripetal force is the force that keeps a particle in uniform circular motion. It acts towards the center of the circle and is necessary to balance the outward centrifugal force that is exerted on the particle due to its motion. Without centripetal force, the particle would move in a straight line tangent to the circle.

4. What is the role of velocity in uniform circular motion?

Velocity is an important factor in uniform circular motion as it determines the speed and direction of the particle. In this type of motion, the velocity vector is always tangent to the circle and its magnitude remains constant. The change in direction of the velocity vector is what causes the particle to move in a circular path.

5. How is angular velocity related to uniform circular motion?

Angular velocity is the rate of change of angular displacement and is closely related to uniform circular motion. It is defined as the angle swept by the particle per unit time and is directly proportional to the linear speed of the particle and the radius of the circular path. In uniform circular motion, the angular velocity remains constant.

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