Rotation and Translation - wheel

[sami23]

Homework Statement

An exhausted bicyclist pedals somewhat erratically. The angular velocity of the front tire, as measured with respect to an axis fixed at the tire’s center, is given by

omega (t)= (1/2)t - (1/4)sin (2t) for t >= 0

where t represents the time in seconds and omega (t) is measured in radians per second. Assume that the tires roll without slipping.

If the tire’s radius is 23 m, what is d, the magnitude of the spot’s displacement after 2.0 seconds?

Homework Equations

d = r * theta (in radians)

The Attempt at a Solution

after integrating omega(t) dt: theta = (1/4)t + (1/8)cos(t) where t =2
theta = 0.7932 rad
d = 23*(0.7932) = 18.24 m
But it's wrong.

The magnitude of the displacement of the spot is the shortest distance traveled by the spot between its initial and final positions. Square and add the magnitudes of the displacement of the spot in the horizontal and vertical directions and take the square root to determine the displacement of the spot.

Please help. I don't get it.

 

[mark726]

it is important to carefully analyze the given problem and approach it in a systematic way. Let's break down the problem and see where the mistake might be.

Firstly, the given equation for angular velocity is only valid for t>=0, so we need to make sure our calculations also fall within this range.

Next, let's look at the equation for displacement, d = r*theta. This equation is only applicable when the object is moving in a circular path. In this case, the bicyclist is moving in a straight line, so we cannot use this equation directly.

However, we can use the concept of arc length to find the displacement. Arc length (s) is defined as s = r*theta, where r is the radius and theta is the central angle in radians. In this case, the central angle is given by the integrated angular velocity, which we have already calculated to be (1/4)t + (1/8)cos(t).

Now, we need to find the arc length traveled by the spot in 2 seconds. To do this, we can use the formula s = r*theta, where r is the radius (23m) and theta is the central angle in radians. So, the arc length traveled by the spot in 2 seconds is:

s = (23m)*[(1/4)*(2s) + (1/8)*cos(2s)]
= (23m)*[0.5 + (1/8)*cos(2)]
= (23m)*[0.5 + (1/8)*(1)] [since cos(2)=1]
= 11.5m + 1.4375m
= 12.9375m

Therefore, the magnitude of the spot's displacement after 2 seconds is approximately 12.9375m.

It is always important to carefully analyze the given information and use the correct equations and concepts to solve a problem. I hope this explanation helps you understand the solution better.