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?

(a) Well, let's hope that Sam doesn't swing so hard that he launches himself into orbit! To find the final angular velocity of the club, we can use the formula:

Final angular velocity = Initial angular velocity + (Angular acceleration × Time)

Here, the initial angular velocity is given as 5 rad/s, the angular acceleration is 100 rad/s^2, and the time is 0.8 s. So:

Final angular velocity = 5 rad/s + (100 rad/s^2 × 0.8 s)

Let me get my calculator... *clickety-clackety*... and the final angular velocity is...

drumroll please...

85 rad/s! Phew, I hope Sam can keep up with that!

(b) Now let's calculate the tangential acceleration of the club at the instant it hits the ball. We can use the formula:

Tangential acceleration = Radius × Angular acceleration

The radius is given as 1.5 m and the angular acceleration is still 100 rad/s^2. So:

Tangential acceleration = 1.5 m × 100 rad/s^2

And the answer is...

150 m/s^2! That club is really accelerating!

(c) Lastly, we need to find the radial acceleration of the club at the instant it hits the ball. The radial acceleration can be calculated using the formula:

Radial acceleration = Radius × (Angular velocity)^2

The radius is 1.5 m, and the final angular velocity is 85 rad/s (which we found in part a). So:

Radial acceleration = 1.5 m × (85 rad/s)^2

And... the magical answer is...

1083.75 m/s^2! Looks like Sam's swing is out of this world! Be careful, Sam, you wouldn't want to send the ball to another dimension!

To find the final angular velocity (ωf) of the club at the instant it hits the ball during the downswing, we can use the angular acceleration (α) and the time (t) for the downswing.

(a) The final angular velocity can be found using the formula:

ωf = ωi + αt

Where:
ωi = Initial angular velocity = 5 rad/s
α = Average angular acceleration = 100 rad/s^2
t = Time for the downswing = 0.8 s

Substituting the values, we get:

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.

(b) Tangential acceleration (at) can be calculated using the formula:

at = αr

Where:
α = Average angular acceleration = 100 rad/s^2
r = Distance from the club head to the axis of rotation = 1.5 m

Substituting the values, we get:

at = (100 rad/s^2)(1.5 m)
=> at = 150 m/s^2

Therefore, the tangential acceleration of the club at the instant it hits the ball during the downswing is 150 m/s^2.

(c) Radial acceleration (ar) can be found using the formula:

ar = rω^2

Where:
r = Distance from the club head to the axis of rotation = 1.5 m
ω = Angular velocity = ωf (final angular velocity)

Substituting the values, we get:

ar = (1.5 m)(85 rad/s)^2
=> ar = (1.5 m)(7225 rad^2/s^2)
=> ar = 10837.5 m/s^2

Hence, the radial acceleration of the club at the instant it hits the ball during the downswing is 10837.5 m/s^2.