6. A solid disk rotates in the horizontal plane at an angular velocity of 4.10 × 10-2 rad/s with respect to an axis perpendicular to the disk at its center. The moment of inertia of the disk is 0.17 kg.m². From above, sand is dropped straight down onto this rotating disk, so that a thin uniform ring of sand is formed at a distance of 0.40 m from the axis. The sand in the ring has a mass of 0.50 kg. After all the sand is in place, what is the angular velocity of the disk? rad/s f60 ssf f60 Jss* 50 ssi
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 6: Angular Velocity of a Rotating Disk with Added Mass
A solid disk rotates in the horizontal plane at an angular velocity of \(4.10 \times 10^{-2}\) rad/s with respect to an axis perpendicular to the disk at its center. The moment of inertia of the disk is 0.17 kg·m². From above, sand is dropped straight down onto this rotating disk, forming a thin uniform ring of sand at a distance of 0.40 m from the axis. The sand in the ring has a mass of 0.50 kg. After all the sand is in place, what is the angular velocity of the disk?
\[ \text{Angular Velocity:} \quad \_\_\_\_ \text{ rad/s} \]
**Explanation:**
To find the new angular velocity after the sand is added, we will use the principle of conservation of angular momentum.
1. **Initial Angular Momentum**:
- Angular momentum (L) = Moment of Inertia (I) × Angular Velocity (ω).
- For the disk: \( L_{\text{initial}} = I_{\text{disk}} \times \omega_{\text{initial}} \).
- Given: \( I_{\text{disk}} = 0.17 \, \text{kg·m}² \)
- Given: \( \omega_{\text{initial}} = 4.10 \times 10^{-2} \, \text{rad/s} \)
- So, \( L_{\text{initial}} = 0.17 \, \text{kg·m}² \times 4.10 \times 10^{-2} \, \text{rad/s} = 6.97 \times 10^{-3} \, \text{kg·m}²/\text{s} \).
2. **Moment of Inertia of the Sand**:
- The moment of inertia of a ring (I_ring) = mass (m) × radius² (r²).
- Given: \( m_{\text{ring}} = 0.50 \, \text{kg} \) and \( r = 0.40 \, \text{m} \)
- \( I_{\text{ring}} = 0.50 \, \text{kg} \times (0.40 \, \text{m})](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c6fb9be-cb3b-43fd-a6cc-90b4ac965f54%2F893e9e10-a8ca-4846-9405-052a0674c4c9%2Fplcy4vn_processed.png&w=3840&q=75)
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