A bucket of mass M = 5.47 kg is attached to a second bucket of mass m = 3:52 kg by an ideal string. The string is hung over an ideal pulley as shown in the �gure. Mass M is started with an initial downward speed of 2.13 m/s. What then is the speed (m/s) of mass M after it has moved 2.47 meters?
To find the speed of mass M after it has moved 2.47 meters, we can use the principle of conservation of mechanical energy.
The mechanical energy of the system is conserved because there are no external forces doing work on it. The mechanical energy is given by the sum of the potential energy and the kinetic energy.
The potential energy of the system can be defined as the sum of the potential energy due to the gravitational forces acting on each of the masses. We can calculate the potential energy of the system at the initial point and at the final point.
The initial potential energy is given by:
PE_initial = M * g * h_initial,
where M is the mass of the large bucket, g is the acceleration due to gravity, and h_initial is the initial height of the large bucket.
The final potential energy is given by:
PE_final = M * g * h_final,
where h_final is the final height of the large bucket after it has moved 2.47 meters.
The kinetic energy is given by:
KE_initial = (1/2) * M * v_initial^2,
where v_initial is the initial downward speed of the large bucket.
The final kinetic energy is given by:
KE_final = (1/2) * M * v_final^2,
where v_final is the speed of the large bucket after it has moved 2.47 meters.
Since the mechanical energy is conserved, the initial mechanical energy (potential energy + kinetic energy) is equal to the final mechanical energy:
PE_initial + KE_initial = PE_final + KE_final.
Substituting the equations for potential and kinetic energy, we have:
M * g * h_initial + (1/2) * M * v_initial^2 = M * g * h_final + (1/2) * M * v_final^2.
Simplifying the equation and solving for v_final, we get:
v_final^2 = v_initial^2 + 2 * g * (h_initial - h_final).
Finally, taking the square root of both sides, we find the speed of mass M after it has moved 2.47 meters:
v_final = √(v_initial^2 + 2 * g * (h_initial - h_final)).
Substituting the known values into the equation, we can calculate the speed of mass M after it has moved 2.47 meters.