Angular Displacement of bike pedals

In summary, the angular displacement of the spot of paint on the front tire of the bicycle from time 0 to 2 seconds, as described by the equation ω(t)=(1/2)t - (1/4)sin(2t), is approximately 0.793 radians. The correct integral is ((1/4)(t^2))+((1/8)(cos2t)) from 0 to 2 seconds.
  • #1
snowmx0090
3
0
An exhausted bicyclist pedals somewhat erratically, so that the angular velocity of his tires follows the equation,
ω(t)=(1/2)t - (1/4)sin(2t)
where represents time (measured in seconds).
There is a spot of paint on the front tire of the bicycle. Take the position of the spot at time to be at angle radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and measure positive angles in the direction of the tire's rotation. What angular displacement has the spot of paint undergone between time 0 and 2 seconds?
Express your answer in radians.

I thought you would take the integral of the equation from t=0 to t=2 in which the equation would then be (1/4)t^2 + (1/2)cos(2t) but this was wrong. Where should I start with this equation?
 
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  • #2
snowmx0090 said:
I thought you would take the integral of the equation from t=0 to t=2 in which the equation would then be (1/4)t^2 + (1/2)cos(2t) but this was wrong.
Check your integration--the second term has an error.
 
  • #3
Would the equation after integrating be (1/4)t^2 + (1/4)cos(2t)? Was this my mistake?
 
  • #4
snowmx0090 said:
Would the equation after integrating be (1/4)t^2 + (1/4)cos(2t)? Was this my mistake?

There is still a mistake. What does [tex]\frac{d}{dt}(\frac{1}{4}\cos (2t))[/tex] equal?
 
  • #5
you would have to use a u substitute. Set u=2t, and du=2. So instead of it being 1/4cos(2t), it would be 1/8cos(2t).

final : ((1/4)(t^2))+((1/8)(cos2t)) from 0 to 2 seconds

.918295 - .125 = .793 radians
 
Last edited:

1. What is angular displacement?

Angular displacement is the measure of the change in angle of an object or system. In the context of a bike, it refers to the change in angle of the bike pedals as they rotate during pedaling.

2. How is angular displacement measured?

Angular displacement is typically measured in radians or degrees. In the case of bike pedals, it can be measured using a protractor or by attaching a sensor to the pedal and measuring the rotation electronically.

3. What factors affect angular displacement of bike pedals?

The angular displacement of bike pedals can be affected by various factors, such as the length of the bike crank, the gear ratio, the force applied by the rider, and the resistance of the bike pedals.

4. How does angular displacement impact bike performance?

The angular displacement of bike pedals plays a crucial role in the performance of the bike. It determines the speed and efficiency of pedaling, which in turn affects the overall speed and distance covered by the bike.

5. What is the relationship between angular displacement and torque?

Angular displacement and torque are directly proportional to each other. This means that as the angular displacement of the bike pedals increases, so does the torque required to rotate them. This relationship is important in understanding the power and force needed to pedal a bike.

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