A short box with a large flat base is made of iron. The box has mass mI = 890 kg and specific
heat cI = 450 J/kg◦C. The box is filled with water. The water has mass mW = 1220 kg and specific
heat cW = 4180 J/kg◦C. The box and water are initially at temperature T0 = 20◦C. The box is
chained to the back of a truck and is dragged down the road. The coefficient of kinetic friction
between the road and the box is μ = 0.70. The fraction of the mechanical energy dissipated by
friction that is added as heat to the box is f = 0.40.
Neglect any loss of heat to the surroundings and determine how far the truck travels before the
water reaches the boiling point.
To determine how far the truck travels before the water reaches the boiling point, we need to find the amount of heat transferred to the water.
The heat transferred to the water can be calculated using the formula:
Q = mcΔT
where Q is the heat transferred, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature.
First, let's calculate the initial temperature difference:
ΔT_initial = T_boiling - T0
where T_boiling is the boiling point of water.
Next, let's find the heat transferred to raise the temperature of the water from T0 to the boiling point:
Q1 = mW * cW * ΔT_initial
Now, we need to calculate the work done by friction. The frictional force can be calculated using the equation:
F_friction = μ * mI * g
where F_friction is the frictional force, μ is the coefficient of kinetic friction, mI is the mass of the box, and g is the acceleration due to gravity.
The work done by friction can be calculated using the formula:
W_friction = F_friction * d
where W_friction is the work done by friction and d is the distance traveled by the truck.
The work done by friction is dissipated as heat in the box, so the amount of heat added to the box is:
Q2 = f * W_friction
Finally, equating the amount of heat received by the water with the heat added to the box, we have:
Q1 = Q2
Substituting the values and solving for d, we can find the distance traveled by the truck before the water reaches the boiling point:
d = Q1 / (f * F_friction)
Let's calculate the distance d.
To determine how far the truck travels before the water reaches the boiling point, we need to calculate the amount of heat transferred to the water due to friction.
1. Calculate the change in temperature of the water:
ΔT = 100°C - 20°C = 80°C
2. Calculate the amount of heat transferred to the water:
Q = mW * cW * ΔT
Q = 1220 kg * 4180 J/kg°C * 80°C
Q = 405,376,000 J
3. Calculate the mechanical energy dissipated by friction:
E_friction = f * initial mechanical energy
E_friction = f * (mass of box * acceleration due to gravity * distance)
E_friction = 0.40 * (890 kg * 9.8 m/s² * distance)
4. Equate the heat transferred to the water to the mechanical energy dissipated by friction:
Q = E_friction
405,376,000 J = 0.40 * (890 kg * 9.8 m/s² * distance)
5. Solve for the distance:
distance = 405,376,000 J / (0.40 * 890 kg * 9.8 m/s²)
distance = 1,080,000 meters (rounded to the nearest meter)
Therefore, the truck travels approximately 1,080,000 meters (or 1,080 kilometers) before the water reaches the boiling point.