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) constant acceleration, like a spring mass or pendulum or missle, comes back through zero at same speed, opposite direction. omega = +5 rad/s if it was -5 rad /s at start.
(b) alpha = d omega/dt = constant = 100 rad/s so alpha R = 1.5 * 100 = 150 m/s^2
(c) Ac = omega^2 R = 25 * 1.5 = 37.5 meters/s^2
To answer these questions, we need to use the equations of rotational motion. Let's break down each part of the problem.
(a) To find the final angular velocity of the club at the instant it hits the ball during the downswing, we can use the equation:
ω₂ = ω₁ + αt
where ω₂ is the final angular velocity, ω₁ is the initial angular velocity, α is the angular acceleration, and t is the time.
Given:
Initial angular velocity (ω₁) = 5 rad/s
Angular acceleration (α) = 100 rad/s²
Time (t) = 0.8 s
Substituting the given values into the equation, we can calculate:
ω₂ = 5 rad/s + (100 rad/s²) * (0.8 s)
ω₂ = 5 rad/s + 80 rad/s
ω₂ = 85 rad/s
Therefore, the final angular velocity of the club is 85 rad/s at the instant it hits the ball during the downswing.
(b) To find the tangential acceleration at the instant the club hits the ball, we can use the formula:
at = rα
where at is the tangential acceleration, r is the radius (distance from the club head to the axis of rotation), and α is the angular acceleration.
Given:
Radius (r) = 1.5 m
Angular acceleration (α) = 100 rad/s²
Plugging in the values into the equation, we can calculate:
at = (1.5 m) * (100 rad/s²)
at = 150 m/s²
Therefore, the tangential acceleration of the club at the instant it hits the ball is 150 m/s².
(c) The radial acceleration at the instant the club hits the ball is given by the formula:
ar = rω²
where ar is the radial acceleration, r is the radius, and ω is the angular velocity.
Given:
Radius (r) = 1.5 m
Angular velocity (ω) = 85 rad/s
Using the equation, we can calculate:
ar = (1.5 m) * (85 rad/s)²
ar = 1826.25 m/s²
Therefore, the radial acceleration of the club at the instant it hits the ball is 1826.25 m/s².