Uniform Circular Motion

Learning Goal:

To find the velocity and acceleration of an object undergoing uniform circular motion.

Suppose that a particle's position is given by the following expression:

r(t)=R[cos(ωt)i+sin(ωt)j]
 =Rcos(ωt)i+Rsin(ωt)j

where ω is the constant value of the particle's angular velocity.

Motion of the particle

How would you describe the particle's motion?

A circle starting at time t=0 on the positive x axis.

Time required by the particle to cross the negative x axis for the first time

When does the particle first cross the negative x axis?

Express your answer in terms of some or all of the variables ω , R , and π .

t−x =

πω

Velocity of the particle

Find the particle's velocity as a function of time, v(t) .

Express your answer using unit vectors (e.g., Ai+Bj , where A and B are functions of some or all of the variables ω , R , t , and π ).

v(t) =

R[−ωsin(ωt)i+ωcos(ωt)j]

Speed of the particle

Find the particle’s speed at time t .

Express your answer in terms of some or all of the variables ω , R , and π .

v(t) =

(Rωsin(ωt))2+(Rωcos(ωt))2−−−−−−−−−−−−−−−−−−−−−−−√

Acceleration of the particle as a function of time

Find the particle’s acceleration as a function of time.

Express your answer using unit vectors (e.g., Ai+Bj , where A and B are functions of some or all the variables ω , R , t , and π ).

R[−ω2cos(ωt)i−ω2sin(ωt)j]

a(t)=R[-w^2cos(wt)i--w^2sin(wt)j]

Acceleration of the particle in terms of position

Your calculation is actually a derivation of the centripetal acceleration. To see this, express the particle’s acceleration in terms of its position, r(t) .

Express your answer in terms of some or all of the variables r(t) and ω .

a(t) =

−ω2revec(t)

Magnitude of the particle's acceleration

Find the acceleration vector’s magnitude as a function of time.

Express your answer in terms of some or all of the variables R , ω , and t

a(t) =

(Rω2cos(ωt))2+(Rω2sin(ωt))2−−−−−−−−−−−−−−−−−−−−−−−−−√

Magnitude of the particle's acceleration in terms of the radius of rotation and the particle’s linear speed

Express the magnitude of the particle's acceleration vector in terms of R and v using the expression you obtained for the particle’s speed (from Part D).

Express your answer in terms of some or all of the variables R and v .

a =

v2R