Uniform Circular Motion:
Suppose that a particle's position is given by the following expression:
r(t) = Rcos(omega*t)i + Rsin(omega*t)j
1. Choose the answer that best completes the following sentence:
The particle's motion at t=0 can be described by ____________.
a. an ellipse starting at time t=0 on the positive x axis
b. an ellipse starting at time t=0 on the positive y axis
c. a circle starting at time t=0 on the positive x axis
d. a circle starting at time t=0 on the positive y axis
2. When does the particle first cross the negative x axis?
Express your answer in terms of some or all of the variables omega, R, and pi.
3. Find the particle's velocity as a function of time.
Express your answer using unit vectors (e.g., A i+ B j, where A and B are functions of omega, R, t, and pi).
4. Find the speed of the particle at time t.
Express your answer in terms of some or all of the variables omega, R, and pi.
2. pi/w
3. -wRsin(wt)ihat +wRcos(wt)jhat
4.R*w
1. a circle starting at time on the positive x axis
it is the function r(t).
1.
circle on x axis
1. To determine the particle's motion at t=0, we can substitute t=0 into the expression for r(t).
r(0) = Rcos(omega*0)i + Rsin(omega*0)j
= Rcos(0)i + Rsin(0)j
= R(1)i + 0j
= Ri
Since the x-component is R and the y-component is 0 at t=0, the particle's motion can be described as a circle starting at time t=0 on the positive x-axis.
Therefore, the correct answer is (c) a circle starting at time t=0 on the positive x axis.
2. To find when the particle first crosses the negative x axis, we need to determine when the x-component of r(t) becomes negative.
Setting the x-component equal to 0:
Rcos(omega*t) = 0
This equation holds true when omega*t = pi/2 or 3pi/2 (since the cosine function equals zero at these angles).
Solving for t:
omega*t = pi/2 or 3pi/2
t = (pi/2) / omega or (3pi/2) / omega
Therefore, the particle first crosses the negative x axis at t = (pi/2) / omega or (3pi/2) / omega.
3. To find the particle's velocity as a function of time, we can differentiate the position function with respect to time.
v(t) = d/dt [Rcos(omega*t)i + Rsin(omega*t)j]
= -Romega*sin(omega*t)i + Romega*cos(omega*t)j
Therefore, the particle's velocity is given by v(t) = -Romega*sin(omega*t)i + Romega*cos(omega*t)j.
4. The speed of the particle at time t is the magnitude of its velocity vector.
|v(t)| = sqrt[(-Romega*sin(omega*t))^2 + (Romega*cos(omega*t))^2]
= sqrt[R^2omega^2*sin^2(omega*t) + R^2omega^2*cos^2(omega*t)]
= sqrt[R^2omega^2(sin^2(omega*t) + cos^2(omega*t))]
= sqrt[R^2omega^2]
Therefore, the speed of the particle at time t is |v(t)| = Romega.