A smooth circular hoop with a radius of 0.400 m is placed flat on the floor. A 0.450-kg particle slides around the inside edge of the hoop. The particle is given an initial speed of 8.50 m/s. After one revolution, its speed has dropped to 4.00 m/s because of friction with the floor.

a) Find the energy transformed from mechanical to internal in the particle—hoop—floor system as a result of friction in one revolution.
b) What is the total number of revolutions the particle makes before stopping? Assume the friction force remains constant during the entire motion.

To solve this problem, we can analyze the energy transformations involved in the system.

a) The energy transformed from mechanical to internal in the particle-hoop-floor system can be calculated by finding the difference in kinetic energy of the particle before and after one revolution.

The initial kinetic energy of the particle is given by:
KE_initial = (1/2) * mass * velocity_initial^2

Substituting the given values:
mass = 0.450 kg
velocity_initial = 8.50 m/s

KE_initial = (1/2) * 0.450 kg * (8.50 m/s)^2

After one revolution, the speed of the particle has dropped to 4.00 m/s. Therefore, the final kinetic energy is:
KE_final = (1/2) * mass * velocity_final^2

Substituting the given values:
velocity_final = 4.00 m/s

KE_final = (1/2) * 0.450 kg * (4.00 m/s)^2

Now, the energy transformed from mechanical to internal can be calculated as the difference between the initial and final kinetic energy:
Energy_transformed = KE_initial - KE_final

b) The total number of revolutions the particle makes before stopping can be determined using the conservation of mechanical energy.

The initial mechanical energy of the system is given by the initial kinetic energy of the particle:

ME_initial = KE_initial

The final mechanical energy of the system can be calculated as the sum of the final kinetic energy of the particle and the potential energy of the particle at the topmost point (considering that the vertical displacement is zero):

ME_final = KE_final + PE

The potential energy at the topmost point is given by:

PE = mass * gravity * height

Since the vertical displacement is zero, the potential energy at the topmost point is zero as well.

Therefore, ME_final = KE_final

Now, to find the total number of revolutions, we consider that the mechanical energy is conserved, so:

ME_initial = ME_final

Substituting the expressions for ME:
KE_initial = KE_final

Solving this equation will give the number of revolutions needed for the particle to stop.

Let's calculate the values step-by-step:

a) Energy transformed from mechanical to internal:

mass = 0.450 kg
velocity_initial = 8.50 m/s
velocity_final = 4.00 m/s

KE_initial = (1/2) * 0.450 kg * (8.50 m/s)^2
KE_final = (1/2) * 0.450 kg * (4.00 m/s)^2
Energy_transformed = KE_initial - KE_final

b) Total number of revolutions:

Solve the equation KE_initial = KE_final for the number of revolutions.

Note: To solve this equation, remember that each revolution corresponds to one complete cycle of the particle's motion. The number of revolutions will be a positive whole number.

Now you can go ahead and calculate the values.

To solve these problems, we need to consider the conservation of energy and the work-energy principle.

a) To find the energy transformed from mechanical to internal in the particle-hoop-floor system, we need to determine the initial and final total mechanical energies.

The initial total mechanical energy (E_i) of the particle-hoop system can be calculated as the sum of kinetic energy and potential energy. Since the particle is sliding along the inside edge of the hoop, its potential energy is zero, and only its kinetic energy contributes. So:

E_i = 1/2 * m * v_i^2

where m is the mass of the particle (0.450 kg) and v_i is the initial speed (8.50 m/s).

Similarly, the final total mechanical energy (E_f) is given by:

E_f = 1/2 * m * v_f^2

where v_f is the final speed (4.00 m/s).

The energy transformed from mechanical to internal in one revolution is the difference between E_i and E_f:

Energy transformed = E_i - E_f

Substituting the values:

Energy transformed = 1/2 * 0.450 kg * (8.50 m/s)^2 - 1/2 * 0.450 kg * (4.00 m/s)^2

Evaluate the expression to find the answer.

b) To find the total number of revolutions the particle makes before stopping, we can use the work-energy principle. The work done by friction is equal to the change in mechanical energy.

Since the friction force remains constant, the work done by the friction force can be determined as:

Work done by friction = Force of friction * Distance * Number of revolutions

The work done by friction is equal to the energy transformed from a), and we can calculate it as:

Work done by friction = Energy transformed

Now, we can rearrange the equation to solve for the number of revolutions:

Number of revolutions = Work done by friction / (Force of friction * Distance)

The force of friction can be calculated by multiplying the coefficient of friction (μ) and the normal force (N). The normal force is equal to the weight of the particle, which is given by:

Normal force = mass * gravity

where gravity is the acceleration due to gravity (9.81 m/s^2).

Substitute the values and evaluate the expression to find the total number of revolutions.

a) Relevant equation: Emec = DU + DK

If there are no nonconservative forces, this equation equals zero. Or in this case, some K is transformed into Eint (heat). So the equation becomes: Emec = Ff.d. (F in Newton, distance in meters and N.m = J).
-> DU + DK = Ff.d

Then, 0,5 m vi² = 0,5 m vf² + Ff.d
What do you need is: Ff.d = 0.5 m vi² - 05. m vf². (/!\ all the expression Ff.d is the answer, do not separate it).

b) You know what is the initial speed,
so you have the Ki = 0.5x0.4x64
Or you know that each turn you loose a certain quantity of energy (answer A).
-> Ki/ansA = ...