The flywheel of a prototype car engine is under test. The angular position θ of the flywheel is given by θ = (3.0rad/s3)t3 and the diameter of the flywheel is 36cm.
(a)Find the distance that a particle on the rim moves during that time interval.
(b) Find the angle θ, in radians and in degree, at times t1 = 3.0s and t2 = 6.0s.
(c) Find the average angular velocity, in rad/s and in rev/min, between t1 = 3s and t2 = 6s.
(d) Find the instantaneous angular velocity at time t1 = 3.0s and t2 = 6.0s.
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To answer these questions, we need to use the given equation for the angular position of the flywheel.
(a) The distance that a particle on the rim moves during a time interval can be found by finding the arc length of the corresponding angle. Since the diameter of the flywheel is given, we can use the formula for arc length:
Arc length = (angle in radians) x (radius)
Arc length = θ x (diameter/2)
Given that the diameter of the flywheel is 36 cm, the radius would be half of that, which is 18 cm or 0.18 m. Substituting these values into the formula, we get:
Arc length = θ x 0.18 m
(b) To find the angle θ at times t1 = 3.0s and t2 = 6.0s, we need to substitute these values into the given equation for θ:
θ(t1) = (3.0 rad/s^3) x (3.0 s)^3
θ(t2) = (3.0 rad/s^3) x (6.0 s)^3
To convert these values into degrees, you can use the fact that 1 radian is equal to 180/π degrees.
(c) The average angular velocity can be found by dividing the change in angular position by the time interval:
Average angular velocity = (change in θ) / (change in t)
Average angular velocity = (θ(t2) - θ(t1)) / (t2 - t1)
To convert the average angular velocity from radians per second to revolutions per minute, you can use the conversion factor: 1 revolution = 2π radians and 1 minute = 60 seconds.
(d) The instantaneous angular velocity at a specific time can be found by taking the derivative of the angular position equation with respect to time:
Instantaneous angular velocity = dθ/dt
To find the instantaneous angular velocity at time t1 = 3.0s and t2 = 6.0s, substitute these values into the derivative of the equation:
Instantaneous angular velocity(t1) = dθ(t1)/dt
Instantaneous angular velocity(t2) = dθ(t2)/dt