Question
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
.
a) Find the the angular position of the object relative to equilibrium at
b) Find the the angular displacement of the object from
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
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:
- The derivative of
is . - The derivative of
is . - 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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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{~
a)
b)
c)
d)
e)
f)
g) ( \vec{\alpha}(2.70 \mathrm{~