Question

Обруч массой m=0,3 кг и радиусом R=0,5 м привели во вращение, сообщив ему энергию вращательного движения 1200 Дж, и опустили на пол так, что его ось вращения оказалась параллельной плоскости пола. Если обруч начал двигаться без проскальзывания, имея кинетическую энергию поступательного движения 200 Дж, то сила трения совершила работу, равную…

Answer 1

Calculation of Work Done by Friction

To calculate the work done by friction, we need to determine the force of friction acting on the hoop and the distance over which the force is applied.

Given: - Mass of the hoop, m = 0.3 kg - Radius of the hoop, R = 0.5 m - Rotational kinetic energy imparted to the hoop, E_rot = 1200 J - Kinetic energy of translational motion, E_trans = 200 J

To find the force of friction, we can use the principle of conservation of energy. The total energy imparted to the hoop is the sum of the rotational and translational kinetic energies:

E_total = E_rot + E_trans

Let's calculate the total energy:

E_total = 1200 J + 200 J = 1400 J

Since the hoop is rolling without slipping, the total energy can be expressed as the sum of the rotational and translational energies:

E_total = (1/2)Iω^2 + (1/2)mv^2

where: - I is the moment of inertia of the hoop - ω is the angular velocity of the hoop - v is the linear velocity of the hoop

The moment of inertia of a hoop rotating about its axis is given by:

I = mR^2

Substituting the values, we have:

1400 J = (1/2)(0.3 kg)(0.5 m)^2ω^2 + (1/2)(0.3 kg)v^2

Simplifying the equation, we get:

1400 J = 0.075ω^2 + 0.15v^2

Since the hoop is rolling without slipping, the linear velocity v can be expressed in terms of the angular velocity ω:

v = Rω

Substituting this into the equation, we have:

1400 J = 0.075ω^2 + 0.15(Rω)^2

Simplifying further:

1400 J = 0.075ω^2 + 0.15(0.5ω)^2

1400 J = 0.075ω^2 + 0.0375ω^2

1400 J = 0.1125ω^2

Solving for ω^2:

ω^2 = (1400 J) / 0.1125

ω^2 ≈ 12444.44

Taking the square root of both sides:

ω ≈ 111.57 rad/s

Now that we have the angular velocity, we can calculate the force of friction using the equation:

f_friction = Iα

where: - α is the angular acceleration of the hoop

The angular acceleration can be calculated using the equation:

α = ω / t

where t is the time taken for the hoop to come to rest.

Unfortunately, the given information does not provide the time taken for the hoop to come to rest. Without this information, we cannot calculate the force of friction or the work done by friction.

Please provide the missing information, and I will be happy to assist you further.