Rotational Kinematics of bicycle breaks

In summary, after braking, the bike takes 1.5 seconds to come to a stop. The angular acceleration of each wheel is -1.91 rad/s^2.
  • #1
tem_osu
12
0
[SOLVED] Rotational Kinematics

Homework Statement


A person is riding a bicycle, and its wheels have an angular velocity of +18.0 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 +13.5 revolutions.

(a) How much time does it take for the bike to come to rest?

(b) What is the angular acceleration of each wheel?

Homework Equations


(a) θ = (ω0 + ω)t

(b) ω = ω0 + αt

The Attempt at a Solution


(a) +13.5 rev = (18.0 rad/s - 0 rad/s)t
t = (+13.5 rev)/(18.0 rad/s) = 0.75 - wrong answer

(b) (ω - ω0)/t = α
(0 rad/s - 18.0 rad/s)/0.75 = α - wrong answer


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Homework Statement


After 10.0 s, a spinning roulette wheel at a casino has slowed down to an angular velocity of +1.70 rad/s. During this time, the wheel has an angular acceleration of -5.05 rad/s^2. Determine the angular displacement of the wheel.

Homework Equations


ω^2 = ω0^2 + 2α(θ − θ0)

The Attempt at a Solution


(+1.70 rad/s)^2 = 0 + 2(-5.05 rad/s^2)θ
θ = [(1.70 rad/s)^2]/2(-5.05 rad/s^2) = -0.286 - wrong answer

------------------------------------------------------------------

Please help me, I don't know what I'm doing wrong.
 
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  • #2
Start with 1). Your equation a) is wrong. This is accelerated motion. The equation should be:
theta=omega_0*t+(1/2)*alpha*t^2. Solve your equation b) for alpha and substitute that into a). Now the only variable is t.
 
  • #3
tem_osu said:

Homework Statement


A person is riding a bicycle, and its wheels have an angular velocity of +18.0 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 +13.5 revolutions.

(a) How much time does it take for the bike to come to rest?

(b) What is the angular acceleration of each wheel?

Homework Equations


(a) θ = (ω0 + ω)t

(b) ω = ω0 + αt

The Attempt at a Solution


(a) +13.5 rev = (18.0 rad/s - 0 rad/s)t
t = (+13.5 rev)/(18.0 rad/s) = 0.75 - wrong answer

The equation you have listed as (a) is only valid for uniform angular velocity. Since the wheel is slowing down, velocity is not constant, so you have to use an equation for uniform acceleration (since the problem states that the wheel comes to a uniform stop).Since the acceleration is uniform, you can assume that [itex]\alpha = \frac{\Delta\omega}{\Delta t}[/itex].
 
Last edited:
  • #4
equation 1: θ = ω0t + 1/2αt^2

equation 2: ω = ω0 + αt
(ω - ω0)/t = α

solve for α from equation 2 & plug into equation 1:
θ = ω0t + 1/2[(ω - ω0)/t]t^2

θ = ω0t + 1/2[(ω - ω0)]t
13.5 = 18.0t + 1/2(0 - 18.0)t
13.5 = 18.0t - 9t
13.5 = 9.0t
t = 1.5 seconds - but it's still wrong

Did I simplified it wrong?
 
  • #5
A 'rev' is not a 'rad'. How many radians is 13.5 revolutions? Other than that, well done.
 
  • #6
(a) 13.5 x 2Pi = 9.0t
t = 9.42 s

(b) (0-18.0 rad/s)/9.42s = -1.91 rad/s^2

Thank you so much.

I'm still confuse about the second problem though. I tried several ways, but I've gotten some weird answers.
 
  • #7
omega_0 in that problem isn't zero. The wheel starts at velocity omega_0, slows down at a rate of -5.05rad/sec^2 for 10 sec. Then it's speed is 1.7rad/sec. Can you figure out the correct value for omega_0. Think about it...
 
  • #8
ω = ω0 + αt
ω - αt = ω0
(1.70 rad/s) - (-5.05 rad/s^2)(10.0 s) = ω0
ω0 = 52.5 rad/s

ω^2 = ω0^2 + 2αθ
(1.70^2) = (52.2^2) + 2(-5.05)θ
θ = [(1.70^2) - (52.2^2)]/(2 x -5.05) = 269.5 rad

GOT IT!
Thank you so much for helping me. ^-^
 

What is rotational kinematics of bicycle brakes?

Rotational kinematics of bicycle brakes refers to the study of the movement and forces involved in the rotation of the brake mechanism on a bicycle.

How do bicycle brakes work?

Bicycle brakes work by applying friction to the rim or disc of the wheel, which slows down the rotation of the wheel and ultimately stops the bike. This is achieved through the use of brake pads that press against the rim or disc, creating friction and slowing down the bike.

What factors affect the effectiveness of bicycle brakes?

The effectiveness of bicycle brakes is affected by several factors, including the type and condition of the brake pads, the condition of the rim or disc, and the force applied to the brake levers.

How do I adjust my bicycle brakes?

To adjust your bicycle brakes, you will need to adjust the position of the brake pads to ensure they are making proper contact with the rim or disc. You may also need to tighten or loosen the brake cable and adjust the tension of the brake spring.

What are the different types of bicycle brakes?

There are several types of bicycle brakes, including rim brakes, disc brakes, and coaster brakes. Each type has its own advantages and disadvantages, and the choice of which to use depends on the specific needs and preferences of the rider.

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