A figure skater is spinning with an angular velocity of +17.5 rad/s. She then comes to a stop over a brief period of time. During this time, her angular displacement is +5.06 rad. Determine (a) her average angular acceleration and (b) the time during which she comes to rest. (a) Number i 129.85 Units s (b) Number i Units
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Physics Problem: Angular Motion of a Figure Skater
**Problem Statement:**
A figure skater is spinning with an angular velocity of +17.5 rad/s. She then comes to a stop over a brief period of time. During this time, her angular displacement is +5.06 rad. Determine:
1. Her average angular acceleration.
2. The time during which she comes to rest.
**Given:**
- Initial angular velocity (ω₀): \( +17.5 \ \mathrm{rad/s} \)
- Final angular velocity (ω): \( 0 \ \mathrm{rad/s} \)
- Angular displacement (θ): \( +5.06 \ \mathrm{rad} \)
**Find:**
(a) The time (t) during which she comes to rest.
(b) The average angular acceleration (α).
**Solution:**
### (a) Time to Come to Rest
Using the equation of motion for angular displacement:
\[ θ = ω₀t + \frac{1}{2} αt² \]
As the skater comes to rest, we have:
\[ 0 = ω₀ + αt \]
Rearranging for time (t):
\[ t = \frac{- ω₀}{α} \]
Now, substituting ω₀ and solving for \( α \):
### (b) Average Angular Acceleration
Using the same equation of motion:
\[ θ = ω₀t + \frac{1}{2} αt² \]
We need to find the average angular acceleration \( α \):
\[ α = \frac{2(θ - ω₀t)}{t²} \]
**Calculation:**
1. Substitute the given values into the equations derived above.
2. Solve for \( t \) and \( α \).
**Units:**
- Time (t): \( \text{s} \)
- Angular acceleration (α): \( \text{rad/s}² \)
### Insert Calculated Values:
(a) Time to come to rest: \( 129.85 \ \text{s} \)
(b) Average angular acceleration: (To be calculated based on the specific method or given values)
These values will allow students to understand the process of determining the time and average angular acceleration for a figure skater coming to a stop from an initial angular velocity and given angular displacement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd00a953f-93ed-4d4f-930e-90e00b049c18%2F1bd17c56-8728-4c17-a2d5-55139a4329c6%2F6xmxfuv_processed.png&w=3840&q=75)
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