A baton twirler tosses a baton into the air with an initial angular velocity of 2 rev/s (revolutions per second). The baton undergoes a constant angular acceleration of -.015 rev/s^2 while in the air, and its angular velocity when she catches it is .8 rev/s. Determine the number of revolutions the baton made while in the air
Yep! You got it!
To determine the number of revolutions the baton made while in the air, we first need to find the time it took for the baton to be caught.
We can use the equation of motion for angular velocity, which relates the initial angular velocity, final angular velocity, angular acceleration, and time:
ωf = ωi + αt
Where:
ωf = Final angular velocity
ωi = Initial angular velocity
α = Angular acceleration
t = Time
Rearranging the equation, we have:
t = (ωf - ωi) / α
Plugging in the given values, we have:
t = (0.8 rev/s - 2 rev/s) / (-0.015 rev/s^2)
= (-1.2 rev/s) / (-0.015 rev/s^2)
= 80 seconds
Now that we have the time, we can determine the number of revolutions the baton made while in the air using another equation:
θ = ωi*t + (1/2)α*t^2
Where:
θ = Angular displacement
ωi = Initial angular velocity
α = Angular acceleration
t = Time
Since the baton starts with an initial angular velocity of 2 rev/s and it undergoes a constant angular acceleration of -0.015 rev/s^2, we can substitute these values into the equation:
θ = (2 rev/s)(80 s) + (1/2)(-0.015 rev/s^2)(80 s)^2
= 160 rev + (-0.6 rev/s^2)(6400 s^2)
= 160 rev + (-3840 rev)
= -3680 rev
The negative sign indicates that the baton made the rotations in the opposite direction, which is still valid.
Therefore, the baton made approximately 3680 revolutions while in the air.