Question
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The angular position of a 0.353 object moving in a circular path is given by

a) Find the the angular position of the object relative to equilibrium at .

b) Find the the angular displacement of the object from to .

c) Provide an expression of the angular velocity of the object as a function of time (do not include units
and the vector notation in the expression)

d) Find the object’s angular velocity at .

e) Find the object’s average angular velocity between and .

f) Provide an expression of the angular acceleration of the object as a function of time (o not include units
and the vector notation in the expression)

g) Find the object’s angular acceleration at .

Ask by Harper Mann. in the United States
Jan 30,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

a)
b)
c)
d)
e)
f)
g)

Solution

Let’s solve the problem step by step.

a) Find the angular position of the object relative to equilibrium at .

The angular position is given by the equation:
Substituting :
Calculating each term:
Now substituting these values back:

b) Find the angular displacement of the object from to .

First, we need to find :
Now, the angular displacement is:

c) Provide an expression of the angular velocity of the object as a function of time.

The angular velocity is the derivative of the angular position :
Calculating the derivative:
  1. The derivative of is .
  2. The derivative of is .
  3. For , using the quotient rule:
Thus, the expression for angular velocity is:

d) Find the object’s angular velocity at .

Substituting into the angular velocity expression:

e) Find the object’s average angular velocity between and .

The average angular velocity is given by:

f) Provide an expression of the angular acceleration of the object as a function of time.

The angular acceleration is the derivative of the angular velocity :
Calculating the derivative:

g) Find the object’s angular acceleration at .

Substituting into the angular acceleration expression:
Calculating :

Summary of Results:

a) ( \vec{\theta

Answered by UpStudy AI and reviewed by a Professional Tutor

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Explain
Simplify this solution

Mind Expander

To find the angular position of the object at , substitute into the given angular position equation:
Calculating this gives:
So, .
For the angular displacement from to , calculate and subtract from :
Thus, the angular displacement :
So, .
Next, the angular velocity as a function of time can be found by differentiating the angular position with respect to time:
Now, at :
Calculating this gives .
To find the average angular velocity between and :
Finally, the angular acceleration as a function of time can be found by differentiating the angular velocity:
At :
In summary:
a)
b)
c)
d)
e)
f)
g) ( \vec{\alpha}(2.70 \mathrm{~

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