A 0.45 kg tetherball is attached to a pole and rotating in a horizontal circle of radius r₁ = 1.4 m and is circling at angular speed = 1.42 rad/s. As the rope wraps around the pole the radius of the circle shortens and became r2 = 0.9 m, that decreases the moment of inertia of the rotating tetherball. Neglecting air resistance what will be the angular speed of the ball after the rope is wrapped around the pole measured in kgm2/s (answer with 3 decimal places)?
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.
![**Problem Statement: Conservation of Angular Momentum in Tetherball**
A 0.45 kg tetherball is attached to a pole and rotating in a horizontal circle of radius \( r_1 = 1.4 \) m and is circling at angular speed \( \omega_1 = 1.42 \) rad/s. As the rope wraps around the pole, the radius of the circle shortens and becomes \( r_2 = 0.9 \) m, which decreases the moment of inertia of the rotating tetherball. Neglecting air resistance, what will be the angular speed of the ball after the rope is wrapped around the pole, measured in kgm²/s (answer with 3 decimal places)?
**Analysis & Solution:**
To solve this problem, we use the principle of conservation of angular momentum. The angular momentum \(L\) of the tetherball must remain constant because there are no external torques acting on it.
The formula for angular momentum \(L\) is given by:
\[ L = I \omega \]
Where:
- \( I \) is the moment of inertia
- \( \omega \) is the angular speed
For a point mass \( m \) rotating at a radius \( r \):
\[ I = m r^2 \]
Initially:
\[ L_1 = I_1 \omega_1 = m r_1^2 \omega_1 \]
After the radius changes:
\[ L_2 = I_2 \omega_2 = m r_2^2 \omega_2 \]
Since angular momentum is conserved:
\[ L_1 = L_2 \]
\[ m r_1^2 \omega_1 = m r_2^2 \omega_2 \]
Solving for \( \omega_2 \):
\[ \omega_2 = \frac{r_1^2 \omega_1}{r_2^2} \]
Substituting the given values:
\[ m = 0.45 \, \text{kg} \]
\[ r_1 = 1.4 \, \text{m} \]
\[ \omega_1 = 1.42 \, \text{rad/s} \]
\[ r_2 = 0.9 \, \text{m} \]
\[ \omega_2 = \frac{(1.4 \, \text{m})^2 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcde4937d-e883-446c-8941-8d2ade4d99b0%2Ff7cda6aa-c49c-4b24-8b0a-2d42bf2bf7cb%2Fo51wpj_processed.jpeg&w=3840&q=75)
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