Sam is at the driving range practicing his swing. During Sam’s golf drive, the initial angular velocity of his club is 5 rad/s at the start of the backswing. The constant, average angular acceleration of the club between the start of the backswing and the instant it makes contact with the ball is 100 rad/s2. The distance from the club head to the axis of rotation is 1.5 m at the instant that the club hits the ball. The downswing lasts 0.8 s.

Question

Sam is at the driving range practicing his swing. During Sam’s golf drive, the initial angular velocity of his club is 5 rad/s at the start of the backswing. The constant, average angular acceleration of the club between the start of the backswing and the instant it makes contact with the ball is 100 rad/s2. The distance from the club head to the axis of rotation is 1.5 m at the instant that the club hits the ball. The downswing lasts 0.8 s.

(a) What is the final angular velocity of the club at
the instant that it hits the ball during the downswing?
(b) What is the tangential acceleration of the
club at the instant it hits the ball during the downswing?
(c) What is the radial acceleration of the
club at the instant it hits the ball during the downswing?

Answer 1

To find the final angular velocity of the club at the moment it hits the ball, we can use the formula for angular velocity:

ωf = ωi + αt

Where:
ωf is the final angular velocity
ωi is the initial angular velocity
α is the angular acceleration
t is the time

Substituting the given values:
ωf = 5 rad/s + (100 rad/s^2)(0.8 s)
ωf = 5 rad/s + 80 rad/s
ωf = 85 rad/s

So the final angular velocity of the club at the instant it hits the ball during the downswing is 85 rad/s.

To find the tangential acceleration of the club at the instant it hits the ball, we can multiply the radial distance of the club head from the axis of rotation (r) by the square of its angular velocity:

Tangential acceleration (at) = r × (ωf)^2

Substituting the given values:
r = 1.5 m
ωf = 85 rad/s

at = (1.5 m) × (85 rad/s)^2
at = 1.5 m × 7225 rad^2/s^2
at = 10,837.5 m/s^2

So the tangential acceleration of the club at the instant it hits the ball during the downswing is 10,837.5 m/s^2.

To find the radial acceleration of the club at the instant it hits the ball, we can multiply the radial distance of the club head from the axis of rotation (r) by the square of its angular velocity:

Radial acceleration (ar) = r × α

Substituting the given values:
r = 1.5 m
α = 100 rad/s^2

ar = (1.5 m) × (100 rad/s^2)
ar = 150 m/s^2

So the radial acceleration of the club at the instant it hits the ball during the downswing is 150 m/s^2.