Question

A string is wrapped around a uniform disk of mass M= 1.3 kg and radius R= 0.11 m (see figure below). Attached to the disk are four low-mass rods of radius b= 0.12 m, each with a small mass m= 0.6 kg at the end. The device is initially at rest on a nearly frictionless surface. Then you pull the string with a constant force F= 20 N for a time of 0.2 s. Now what is the angular speed of the apparatus?

rad/s
Also calculate numerically the angle through which the apparatus turns, in radians and degrees.
Δ⁢θradians= rad
Δ⁢θdegrees= °

Answer 1

To find the angular speed of the apparatus, calculate the final angular momentum of the system using the moment of inertia of the disk and rods. Then, solve for the angular speed using the equation for angular momentum. To calculate the angle through which the apparatus turns, use the formula Δθ = w * t. Convert the angle to radians and degrees using the appropriate formulas.

Step-by-step explanation:

To find the angular speed of the apparatus, we can use the principle of conservation of angular momentum. The initial angular momentum of the system is zero since it's at rest. The final angular momentum is given by the sum of the angular momentum of the disk and the rods. The angular momentum of a rotating object is given by the product of its moment of inertia and angular velocity. So we have:

Lfinal = Idisk * wdisk + 4 * Irod * wrod

Where Idisk is the moment of inertia of the disk, wdisk is the angular speed of the disk, Irod is the moment of inertia of each rod, and wrod is the angular speed of each rod.

Since we are given the mass and radius of the disk, we can calculate its moment of inertia using the formula I = (1/2) * M * R2. We can also calculate the moment of inertia of each rod using the formula I = (1/2) * m * b2, where m is the mass of each rod and b is its radius.

With these values, we can substitute them into the equation for angular momentum and solve for wdisk.

To calculate the angle through which the apparatus turns, we can use the formula Δθ = w * t, where Δθ is the angle, w is the angular speed, and t is the time.

Plugging in the values for wdisk and t, we can calculate the angle in radians. To convert it to degrees, we can use the formula Δθ(degrees) = Δθ(radians) * (180/π).

Answer 2

To find the angular speed of the apparatus, use the principle of conservation of angular momentum. The angular speed is calculated using the equation (FΔt) / (MD²/2 + 4mr²). The angle through which the apparatus turns is found using the equation Δθ = ωΔt. The angle can be converted from radians to degrees using the formula Δθ(degrees) = Δθ(radians) * (180/π).

Step-by-step explanation:

To find the angular speed of the apparatus, we can use the principle of conservation of angular momentum. The initial angular momentum of the apparatus is zero, and when the string is pulled, angular momentum is generated. The equation for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.

Given that the force applied to the string is constant and the time is known, we can use the equation FΔt = ΔL to calculate the change in angular momentum. Since the moment of inertia of the system is the sum of the disk's moment of inertia and the four rods' moment of inertia, we can calculate the angular speed by rearranging the equation.

The angular speed of the apparatus is ω = ΔL / I = (FΔt) / (MD²/2 + 4mr²) = (20 N * 0.2 s) / (1.3 kg * 0.11 m² / 2 + 4 * 0.6 kg * 0.12 m²).

The angle through which the apparatus turns can be found using the equation Δθ = ωΔt. Since the angular speed is now known, the angle Δθ can be calculated by substituting the values of ω and Δt into the formula.

Finally, to convert the angle from radians to degrees, we can use the formula Δθ(degrees) = Δθ(radians) * (180/π).