A person is riding a bicycle, and its wheels have an angular velocity of 26.0 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is 17.6 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the anguar acceleration (in rad/s2) of each wheel?

Question

A person is riding a bicycle, and its wheels have an angular velocity of 26.0 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is 17.6 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the anguar acceleration (in rad/s2) of each wheel?

Do I need to convert revolutions to radians first?

Answer 1

Equations for decelerated motion are

ω=ωₒ - ε•t,
φ = ωₒ•t - ε•t²/2.

Since ω =0, φ = 2•π•N, where N =17.6 rev.
these equatuions are
0=ωₒ-ε•t, …………………… (1)
2•π•N = ωₒ•t - ε•t²/2 …………(2)
From (1)
ε = ωₒ/t……………………… (3)
Plug it in (2) and obtain
t= 4•π•N/ ωₒ.
Then determine angular acceleration using (3).

Answer 2

so first I have to do equation 2? So

2*pi*17.6 = I am not sure what wo is. Would it be the 26.0?

Answer 3

Yes, in order to calculate the time taken for the bike to come to rest and the angular acceleration of each wheel, you will need to convert the given angular displacement from revolutions to radians.

To convert from revolutions to radians, you can use the conversion factor:

1 revolution = 2π radians

So, in this case, where the angular displacement is given in revolutions, you can convert it to radians by multiplying it by 2π.

Now, let's solve the problem.

(a) To find the time taken for the bike to come to rest, you can use the formula:

Time = Angular displacement / Angular velocity

The given angular displacement is 17.6 revolutions, which we can convert to radians by multiplying it by 2π:

Angular displacement in radians = 17.6 revolutions * 2π radians/revolution

(b) To find the angular acceleration of each wheel, you can use the formula:

Angular acceleration = (Final angular velocity - Initial angular velocity) / Time

The final angular velocity is 0 rad/s since the bike comes to a stop. The initial angular velocity is given as 26.0 rad/s.

Now, let's calculate the answers:

(a) Convert revolutions to radians:
Angular displacement in radians = 17.6 revolutions * 2π radians/revolution

(b) Angular acceleration:
Angular acceleration = (0 rad/s - 26.0 rad/s) / Time

So, to answer your question, yes, you need to convert the given angular displacement from revolutions to radians in order to solve this problem.