A 0.349 kg particle slides around a horizontal track. The track has a smooth, vertical outer wall forming a circle with a radius of 1.68 m. The particle is given an initial speed of 8.66 m/s. After one revolution, its speed has dropped to 6.05 m/s because of friction with the rough floor of the track.
A)Calculate the energy loss due to friction in one revolution.
B)Calculate the coefficient of kinetic friction.
C)What is the total number of revolutions the particle makes before stopping?
To find the answers to the given questions, we will need to use a combination of the concept of energy conservation and the laws of circular motion.
A) To calculate the energy loss due to friction in one revolution, we can use the principle of conservation of mechanical energy. The total mechanical energy of the particle can be calculated as the sum of its kinetic energy and potential energy.
The initial mechanical energy of the particle is given by:
E_initial = 1/2 * m * v_initial^2 + m * g * h,
where m is the mass of the particle (0.349 kg), v_initial is the initial speed (8.66 m/s), g is the acceleration due to gravity (9.8 m/s^2), and h is the height from the ground level (0 in this case, as the ground is considered as the reference level).
The final mechanical energy of the particle is given by:
E_final = 1/2 * m * v_final^2 + m * g * h,
where v_final is the final speed (6.05 m/s).
The energy loss due to friction in one revolution is the difference between the initial and final mechanical energies:
ΔE = E_initial - E_final.
Plugging in the given values, we can calculate ΔE:
ΔE = [1/2 * (0.349 kg) * (8.66 m/s)^2] - [1/2 * (0.349 kg) * (6.05 m/s)^2].
B) To calculate the coefficient of kinetic friction, we can use the formula:
f_kinetic = μ_kinetic * N,
where f_kinetic is the force of kinetic friction, μ_kinetic is the coefficient of kinetic friction, and N is the normal force.
In this case, the normal force N is equal to the weight of the particle, which is given by:
N = m * g.
The force of kinetic friction can also be expressed as:
f_kinetic = m * a,
where a is the centripetal acceleration of the particle.
The centripetal acceleration, a, can be calculated using the formula:
a = v^2 / r,
where v is the speed of the particle (6.05 m/s) and r is the radius of the circle (1.68 m).
Now, equating the two expressions for f_kinetic, we have:
μ_kinetic * N = m * a,
μ_kinetic * m * g = m * v^2 / r.
Simplifying the equation, we find:
μ_kinetic = (v^2 / g) / r.
C) To find the total number of revolutions the particle makes before stopping, we can use the equation:
N = (initial speed - final speed) / (2π * radius),
where N is the total number of revolutions.
Plugging in the given values, we can calculate N.
Note: Please provide the values for v_initial and v_final accurately to obtain precise answers.