Calculating Linear Momentum of Golf Club on Impact

[bionut]

(Q) a 108 cm, 0.73kg golf club is swung for 0.5s with a constant acceleartion of 10 rad/s (squ). What os the linear momentum of the club head when it impacts the ball?

Known:
H= 108cm = 1.8m
m = 0.73 kg =7.16N
t = 0.5s
$\alpha$= 10 rad/s (squ)

I know momentum = mass x velcoity

So velcotiy = d/t... (so i need to find the dispalcement)

But I am thorwn but the angular accelertion o 10 rad/s(squ)
The only thing I can think of is using 2(pi)r with r being the 1.8m?

Any sugegstion where to go from here ??

 

[BobG]

If you were talking about linear motion, you could find your linear velocity using your linear acceleration.

$$v_f=v_i + at$$

Same principles apply to rotational motion, except now you're looking at:

$$\omega_f = \omega_i + \alpha t$$

with omega your angular velocity
and alpha your angular acceleration

 

[bionut]

Hi.. thanks what you have said makes sense...

But is I use:

wf=wi + at

wf= 0 + 10 X 1.5
wf = 5 rad/s (sqR)

The linear momentum = mv = 0.73 x 5 = 3.65 kg. m/s

But the answer is 3.9 kg. m/s, I know I am only .25 off but is the book wrong or my calculations wrong ??

 

[BobG]

Your angular velocity is 5 radians/sec.

You need to covert that to linear velocity.

If your unit of measure is your radius (radians is the plural of radius) then you have to multiply by the radius to get your linear velocity. Your final answer was only close because your radius (the club shaft) was close to 1 - it was 1.08 meters.

 

[rwisz]

I would approach this problem by first converting the angular acceleration into linear acceleration. This can be done using the equation a = r * alpha, where a is the linear acceleration, r is the distance from the center of rotation (in this case, the length of the golf club), and alpha is the angular acceleration.

So, a = 1.8m * 10 rad/s^2 = 18 m/s^2

Next, I would use the equation v = u + at to find the final velocity of the club head just before impact. Since the initial velocity (u) is 0, the final velocity (v) will be equal to the velocity gained during the 0.5 seconds of acceleration.

v = 0 + 18 m/s^2 * 0.5s = 9 m/s

Finally, I would calculate the linear momentum using the equation p = mv, where p is the momentum, m is the mass of the club, and v is the final velocity.

p = 0.73 kg * 9 m/s = 6.57 kg*m/s

Therefore, the linear momentum of the golf club at the moment of impact with the ball is 6.57 kg*m/s.