A spool of thin wire pivots on a shaft with friction. The wire is pulled by a hanging mass M=1.50 kg. Upon starting from rest and falling a distance D=1.65 m, the mass has a speed V=2.00 m/s. Calculate the energy lost to friction during that time.
Inner radius: r=0.50 m.
Outer radius: R=0.65 m.
Moment of inertia: Icm=1.09 kg·m2.
To calculate the energy lost to friction, we need to find the initial potential energy, the final kinetic energy, and the work done by the force of friction.
1. Calculate the initial potential energy:
The initial potential energy can be calculated as:
PE_initial = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the height.
Given:
Mass, M = 1.50 kg
Height, D = 1.65 m
Acceleration due to gravity, g = 9.8 m/s^2
So, PE_initial = 1.5 kg * 9.8 m/s^2 * 1.65 m = 24.185 J (Joules)
2. Calculate the final kinetic energy:
The final kinetic energy can be calculated as:
KE_final = 0.5 * m * v^2
where m is the mass and v is the final velocity.
Given:
Mass, M = 1.50 kg
Final velocity, V = 2.00 m/s
So, KE_final = 0.5 * 1.5 kg * (2.0 m/s)^2 = 3.0 J (Joules)
3. Calculate the work done by the force of friction:
The work done by the force of friction can be calculated as:
Work_friction = ∆KE + ∆PE
where ∆KE is the change in kinetic energy and ∆PE is the change in potential energy.
Since the mass is falling and the wire pivots, there is no change in potential energy (∆PE = 0). So, the work done by friction is equal to the change in kinetic energy (∆KE).
Work_friction = KE_final - KE_initial
= 3.0 J - 24.185 J
Therefore, the energy lost to friction during that time is:
Work_friction ≈ -21.185 J (Joules)
Note: The negative sign indicates that energy is lost to friction.