Two blocks with masses m1 = 3.9 kg and m2 = 6.7 kg are connected by a string that hangs over a pulley of mass M = 2.2 kg and radius R = 0.11 m as shown above. The string does not slip. Assuming the system starts from rest, use energy principle find the speed of m2 after it has fallen by 0.4 m. Treat the pulley as a disk.
To find the speed of m2 after it has fallen by 0.4 m using the energy principle, we'll need to consider the potential energy and kinetic energy of the system.
First, let's calculate the potential energy of the system at the initial position. The potential energy is given by the equation:
PE = mgh,
where m is the mass, g is the acceleration due to gravity, and h is the height.
For m1, the initial height is zero, so the potential energy of m1 is zero.
For m2, the initial height is 0.4 m (as it has fallen by 0.4 m), so the potential energy of m2 is:
PE2_initial = m2 * g * h,
= 6.7 * 9.8 * 0.4,
= 26.296 J.
Next, let's calculate the potential energy at the final position (when m2 has fallen by 0.4 m). The potential energy of m2 at the final position is zero since it has reached the ground.
Now, let's consider the kinetic energy of the system. The kinetic energy is given by the equation:
KE = 1/2 * m * v^2,
where m is the mass and v is the velocity.
Since the system starts from rest, the initial kinetic energy is zero.
The final kinetic energy is:
KE_final = 1/2 * m2 * v^2,
= 1/2 * 6.7 * v^2.
According to the conservation of energy principle, the total initial energy is equal to the total final energy. Therefore, we can equate the initial potential energy to the final kinetic energy:
PE2_initial = KE_final.
Substituting the values:
26.296 = 1/2 * 6.7 * v^2.
Now, solve for v:
v^2 = (2 * 26.296) / 6.7,
v^2 = 9.8,
v = √(9.8),
v ≈ 3.13 m/s.
Therefore, the speed of m2 after it has fallen by 0.4 m is approximately 3.13 m/s.