Question

Шар массой m=2 кг и радиусом R=05 м вращается с частотой n=2с в степени -1 вокруг оси, проходящей через его центр. Какую работу надо совершить, чтобы увеличить угловую скорость шара вдвое?

Answer 1

Медный шар радиусом R = 10 см вращается с частотой n = 2 с–1 вокруг оси, проходящей через его центр. Какую работу надо совершить, чтобы увеличить угловую скорость вращения шара вдвое?
А = 35,3 Дж

Answer 2

Calculation of Work to Double the Angular Velocity of a Rotating Sphere

To calculate the work required to double the angular velocity of a rotating sphere, we can use the formula for rotational kinetic energy:

Rotational Kinetic Energy (K) = (1/2) * I * ω^2

Where: - K is the rotational kinetic energy - I is the moment of inertia of the sphere - ω is the angular velocity of the sphere

To double the angular velocity, we need to find the work done to increase the rotational kinetic energy from K to 2K.

Moment of Inertia of a Sphere

The moment of inertia of a solid sphere rotating about its central axis is given by the formula:

I = (2/5) * m * R^2

Where: - I is the moment of inertia - m is the mass of the sphere - R is the radius of the sphere

Given that the mass of the sphere is 2 kg and the radius is 0.5 m, we can calculate the moment of inertia.

I = (2/5) * 2 kg * (0.5 m)^2

Calculating the moment of inertia gives us:

I = 0.4 kg * m^2

Calculation of Work

To calculate the work required to double the angular velocity, we need to find the initial and final angular velocities.

Given that the initial angular velocity is n = 2 s^-1, we can calculate the initial rotational kinetic energy (K_initial) using the formula:

K_initial = (1/2) * I * ω_initial^2

Substituting the values, we get:

K_initial = (1/2) * 0.4 kg * m^2 * (2 s^-1)^2

Simplifying the equation gives us:

K_initial = 0.8 kg * m^2 * s^-2

To double the angular velocity, the final angular velocity (ω_final) will be 2n = 4 s^-1.

We can calculate the final rotational kinetic energy (K_final) using the formula:

K_final = (1/2) * I * ω_final^2

Substituting the values, we get:

K_final = (1/2) * 0.4 kg * m^2 * (4 s^-1)^2

Simplifying the equation gives us:

K_final = 3.2 kg * m^2 * s^-2

The work done to increase the rotational kinetic energy from K_initial to K_final is given by the difference:

Work = K_final - K_initial

Substituting the values, we get:

Work = 3.2 kg * m^2 * s^-2 - 0.8 kg * m^2 * s^-2

Simplifying the equation gives us:

Work = 2.4 kg * m^2 * s^-2

Therefore, the work required to double the angular velocity of the rotating sphere is 2.4 kg * m^2 * s^-2.

Please note that the calculations provided are based on the given information and assumptions.