A bicycle is turned upside down while its owner repairs a flat tire. A friend spins the other wheel, of radius 0.363 m, and observes that drops of water fly off tangentially. She measures the height reached by drops moving vertically (Fig. P10.63). A drop that breaks loose from the tire on one turn rises h = 53.0 cm above the tangent point. A drop that breaks loose on the next turn rises 51.0 cm above the tangent point. The height to which the drops rise decreases because the angular speed of the wheel decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.
v1=√(2gh)=√(2*9.8*0.53)=3.223 m/s
v1=√(2gh)=√(2*9.8*0.53)=3.162 m/s
Let ω=angular velocity
r=0.363 m
ω1=v1/r=8.879 rad/s
ω2=v2/r=8.710 rad/s
Time for one turn between t1 & t2
Δt=2π/((8.879+8.710)/2)
= 0.7145s
Angular acceleration
=(ω2-ω1)/Δt
= -0.237 rad/s²
To determine the magnitude of the average angular acceleration of the wheel, we can use the following steps:
Step 1: Calculate the initial and final angular velocities.
The angular velocity can be calculated using the following formula:
ω = v/r
where ω is the angular velocity, v is the linear velocity, and r is the radius of the wheel.
Given that the height reached by the drops moving vertically on the first turn is 53.0 cm and on the next turn is 51.0 cm, we can conclude that the drop took more time to reach the same height on the second turn. This indicates that the angular velocity decreased.
Step 2: Calculate the change in angular velocity.
The change in angular velocity can be calculated using the following formula:
Δω = ωf - ωi
where Δω is the change in angular velocity, ωf is the final angular velocity, and ωi is the initial angular velocity.
Since the height to which the drops rise decreases because the angular speed of the wheel decreases, we can conclude that the final angular velocity is lower than the initial angular velocity.
Step 3: Calculate the time taken between the two turns.
The time taken between the two turns can be calculated using the following formula:
t = h / (vf - vi)
where t is the time taken, h is the height reached by the drops, vf is the final velocity, and vi is the initial velocity.
Given that the height reached by the drops on the first turn is 53.0 cm and on the next turn is 51.0 cm, we can calculate the time taken.
Step 4: Calculate the average angular acceleration.
The average angular acceleration can be calculated using the following formula:
α = Δω / t
where α is the average angular acceleration, Δω is the change in angular velocity, and t is the time taken.
By plugging in the values calculated in the previous steps, we can determine the magnitude of the average angular acceleration of the wheel.
Note: Ensure that all the units used are consistent. In this case, convert the height measurements to meters and the radius to meters.