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 a certain distance, we need to apply the law of conservation of mechanical energy.
The law of conservation of mechanical energy states that the total mechanical energy of a system remains constant if there are no external forces acting on it.
In this case, the only external force acting on the system is gravity. The potential energy due to gravity is given by the formula:
PE = mgh
Where m is the mass, g is the acceleration due to gravity, and h is the height.
When the mass M moves downward by a distance d, the potential energy of mass M will decrease and transfer to kinetic energy. The equation for kinetic energy is given by:
KE = (1/2)mv^2
Where m is the mass and v is the velocity.
Now, let's find the potential energy of mass M at the start position and the potential energy at the end position:
PE_start = mgh
PE_end = mg(h + d)
The change in potential energy is given by:
ΔPE = PE_end - PE_start
The change in potential energy should be equal to the change in kinetic energy:
ΔPE = ΔKE
Therefore, we can write:
(M + m)g(d + h) - Mgh = (1/2)Mv^2
Now, let's solve for v, which is the velocity of mass M after it has moved a distance of d:
v = √(2g(M + m)d / M)
Substituting the given values:
v = √(2 * 9.8 * (5.47 + 3.52) * 2.47 / 5.47)
v ≈ 4.26 m/s
Therefore, the speed of mass M after it has moved 2.47 meters is approximately 4.26 m/s.