a person riding a bicycle whose wheels have an angular velocity of 20 rad/s. The brakes are applied, and the bike is brought to a stop through a uniform acceleration. During braking, the angular displacement of each wheel is 15.92 revolutions. How long does it take for the bike to come to a stop?
For this question I got 5 seconds but my teacher said it was ten seconds. I do not think I am doing anything wrong. Could someone tell me if I am wrong or right?
during the stop, which starts at 20 and ends at 0, the average angular velocity is 10 rad/s
1 rev = 2pi rad
15.92 revs = 2*pi*15.92 rad
so
2 pi * 15.92 rad/10 rad/s = about 10 s, yes I agree with teacher
Would I use this equation: w=deltaOMEGA/deltaTIME
To solve this problem, we need to use the kinematic equations of motion for rotational motion.
First, let's convert the angular displacement from revolutions to radians:
Given: Angular displacement = 15.92 revolutions
1 revolution = 2π radians
Therefore, angular displacement = 15.92 revolutions * 2π radians/revolution = 100π radians.
Next, we need to find the initial angular velocity and final angular velocity. We are given that the initial angular velocity is 20 rad/s and the final angular velocity is 0 rad/s since the bike comes to a stop.
Next, we can use the following kinematic equation, which relates angular displacement, initial angular velocity, final angular velocity, and time:
θ = ω_i * t + (1/2)α * t^2,
where θ is angular displacement, ω_i is the initial angular velocity, α is the angular acceleration, and t is the time.
Since the bike is coming to a stop, we can assume constant angular acceleration. Therefore, α will be the same for both wheels.
Now, rearrange the equation to solve for time:
θ = ω_i * t + (1/2)α * t^2
100π radians = 20 rad/s * t + (1/2)α * t^2 [Substituting the given values]
Assuming that the braking force acts on both wheels in opposite directions, the total angular acceleration can be calculated as:
α = (ω_f - ω_i)/t
α = (0 rad/s - 20 rad/s)/t
α = -20 rad/s^2/t
Substituting this value of α in the equation:
100π radians = 20 rad/s * t + (1/2) * (-20 rad/s^2/t) * t^2
100π = 20t - 10t
100π = 10t [cancel out common terms]
t = 10π seconds.
Thus, the correct answer is 10π seconds or approximately 31.42 seconds.
Since your answer of 5 seconds is drastically different from the correct answer of 10π seconds or approximately 31.42 seconds, it is recommended to re-check your calculations and equations to identify any errors.