Please Help. I am not sure how to set up or do this problem. Thanks
A car is traveling along a road, and its engine is turning over with an angular velocity of +199 rad/s. The driver steps on the accelerator, and in a time of 14.2 s the angular velocity increases to +263 rad/s.
(a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of +199 rad/s during the entire 14.2-s interval?
(b) What would have been the angular displacement if the angular velocity had been equal to its final value of +263 rad/s during the entire 14.2-s interval?
(c) Determine the actual value of the angular displacement during the 14.2-s interval.
First you multiply 199 with 14.2 to get a.
Then you multiply 263 with 14.2 to get b.
I haven't figured out c yet but i'll let you know!
For c, you find the average of 199 and 263, then you multiply it by 14.2! Hope that helps!
To solve this problem, we need to use the equation that relates angular displacement, angular velocity, and time. The equation is:
θ = ωi * t + (1/2) * α * t^2
where:
θ: angular displacement
ωi: initial angular velocity
t: time
α: angular acceleration (which can be calculated as the change in angular velocity divided by the change in time)
Let's solve each part of the problem step by step:
(a) To find the angular displacement if the angular velocity remained constant at the initial value, we can set α (angular acceleration) equal to zero in the equation. Therefore, the equation simplifies to:
θ = ωi * t
Substituting the given values: ωi = +199 rad/s and t = 14.2 s, we can calculate the angular displacement θ.
θ = (+199 rad/s) * (14.2 s) = +2825.8 rad
(b) To find the angular displacement if the angular velocity remained constant at the final value, we can repeat the same procedure as in part (a) but use the final angular velocity ωf = +263 rad/s.
θ = ωf * t = (+263 rad/s) * (14.2 s) = +3740.6 rad
(c) To determine the actual value of the angular displacement during the 14.2-s interval, we need to use the full equation mentioned at the beginning, which takes into account the change in angular velocity.
First, let's find the angular acceleration (α).
α = (ωf - ωi) / t = (+263 rad/s - +199 rad/s) / (14.2 s) = +4.507 rad/s^2
Now, substitute the values in the equation:
θ = ωi * t + (1/2) * α * t^2
θ = (+199 rad/s) * (14.2 s) + (1/2) * (+4.507 rad/s^2) * (14.2 s)^2
θ = +2825.8 rad + 4528.2 rad
θ = +7354 rad
Therefore, the actual value of the angular displacement during the 14.2-s interval is +7354 rad.
Remember to pay attention to the sign conventions when solving angular displacement problems. In this case, a positive displacement means a counterclockwise rotation and a negative displacement means a clockwise rotation.