Physics rotation question

Question

A flywheel of radius .30 m starts from rest and accelerates with constant angular acceleration of .50 rad/s2. Compute the tangential acceleration, the radial acceleration and the resultant acceleration (magnitude and angle the resultant acceleration makes with the radial direction) of a point on its rim at the following times.

(a) at the start of the motion

(b) just after the flywheel has turned through 120°

Answer 1

(a) Start of the motion or t = 0,

At this instance, the flywheel is at rest. All velocities, linear or angular are zero everywhere. The angular acceleration α is non-zero. So we get

radial acceleration of a point on the rim = - ω² r (positive direction is radially outward from the center) = 0 (rest)

tangential acceleration of a point on the rim = α x r (perpendicular to the radius vector at the point on the rim).

By the way, the angular acceleration α and radius vector to any point on the rim r are always perpendicular in this problem. So the tangential acceleration is always perpendicular to r

resultant acceleration = sum of previous 2 quantities = α x r = 90 degrees to the radius vector r.

(b) flywheel rotates through 120 degrees or 2π/3 radians. The time T it takes to do that is related to the magnitude of α through:

θ = α T² / 2.

From T, the magnitude of the angular velocity at time T can be computed via:

ω = α T.

The various accelerations can now be computed as follows

radial = - ω² r (non-zero unlike in part (a), directed inward toward the center)

tangential = α x r (magnitude is just the product of values of α and r, direction is perpendicular to r in the direction wheel is turning, same as part (a))

resultant = -ω² r + α x r

The resultant acceleration in part (b) is better computed by drawing the vector diagram. The radial and tangential accelerations are legs of a right triangle. The magnitude is length of hypotenuse. The direction can be computed using an appropriate trig ratio e.g.

tan γ = (magnitude of tangential) / (magnitude of radial)

where γ is the angle of the resultant with the radial.

Answer 2

The radial (sometimes called centripetal) acceleration is the acceleration needed for an object to move in a curved path - such as a circle for a flywheel. It's larger for an object moving fast, and is smaller for a larger (more gentle) circle path. The exact equation is

a_radial = v²/r

From math, tangential velocity and acceleration increase with radius.

a_T = r*a_angular

_vT = r*v_angular

At the start, the velocity is obviously zero. So your task is to take the fact that the flywheel begins at rest and spins up with a constant, know, angular acceleration and find the angular velocity after rotating 120°.

Note: Your angular acceleration uses radians for angle, but the final angle is in degrees. You'll need to either convert the 120° into radians or the angular acceleration into degrees/s².