A 6 kg mass is attached to a light cord that passes over a massless, frictionless pulley. The other end of the cord is attached to a 3.5 kg mass as in Figure below. Use conservation of energy to determine the final speed of the 6kg mass after it has fallen (starting from rest) 2m.
To determine the final speed of the 6 kg mass, we can use the principle of conservation of energy. This principle states that the total energy of a system remains constant if no external forces are acting on it.
In this case, the initial energy of the system is solely gravitational potential energy, and the final energy is a combination of gravitational potential energy and kinetic energy.
Step 1: Calculate the initial gravitational potential energy.
The initial gravitational potential energy (U_i) of the 6 kg mass is given by:
U_i = mgh
where m is the mass (6 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height it falls (2 m).
U_i = (6 kg)(9.8 m/s²)(2 m)
Step 2: Calculate the final gravitational potential energy.
The final gravitational potential energy (U_f) is given by:
U_f = mgh
where m is the mass (6 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the final height (0 m since it falls to the ground).
U_f = (6 kg)(9.8 m/s²)(0 m) = 0 J
Step 3: Calculate the change in gravitational potential energy.
The change in gravitational potential energy (ΔU) is given by:
ΔU = U_f - U_i
ΔU = 0 J - [(6 kg)(9.8 m/s²)(2 m)]
Step 4: Calculate the final kinetic energy.
The final kinetic energy (K_f) is equal to the change in gravitational potential energy:
K_f = ΔU
Step 5: Calculate the final speed.
The final speed (v_f) can be calculated using the formula for kinetic energy:
K_f = 1/2mv_f²
Where m is the mass (6 kg) and v_f is the final speed.
1/2mv_f² = ΔU
Step 6: Solve for the final speed.
v_f = √(2ΔU / m)
Now substitute the values into the formula and calculate the final speed:
v_f = √[(2ΔU) / m] = √[(2 × [(6 kg)(9.8 m/s²)(2 m)]) / (6 kg)]
Simplifying the equation will give you the final speed of the 6 kg mass after it has fallen 2 m.