a farmer drives a 0.100 kg iron spike with a 2.00 kg sledge hammer. the sledge hammer moves at a speed of 3.00 m/s and comes to rest on the spike after each swing. assuming all the energy is absorbed by the nail and ignoring the work by the nail, how much would the nails temperature rise after 10 successive swings ?
To determine the temperature rise of the nail after 10 successive swings, we need to calculate the total energy transferred from the sledgehammer to the nail.
First, we can calculate the initial kinetic energy of the sledgehammer before it strikes the nail. The kinetic energy (KE) can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Where:
mass = 2.00 kg (mass of the sledgehammer)
velocity = 3.00 m/s (speed of the sledgehammer)
Plugging in the values, we get:
KE = (1/2) * 2.00 kg * (3.00 m/s)^2 = 9.00 Joules
Since the hammer comes to rest after each swing, all the kinetic energy is transferred to the nail. Therefore, the energy transferred to the nail with each swing is 9.00 Joules.
Now, we can calculate the total energy transferred to the nail after 10 swings by multiplying the energy transferred per swing by the number of swings:
Total energy = Energy per swing * Number of swings
Total energy = 9.00 Joules * 10 = 90.00 Joules
Finally, to determine the temperature rise of the nail, we need to know the heat capacity of the nail. The specific heat capacity (c) of iron is approximately 450 J/kg°C.
Since the mass of the nail is not given, we cannot directly calculate the temperature rise. We need to know the mass of the nail to proceed with further calculations.
To calculate the temperature rise of the nail, we need to use the principle of conservation of energy.
The initial kinetic energy of the sledgehammer can be calculated using the formula:
KE(initial) = 0.5 * mass * velocity^2
Given:
Mass of sledgehammer (m1) = 2.00 kg
Speed of sledgehammer (v1) = 3.00 m/s
KE(initial) = 0.5 * 2.00 kg * (3.00 m/s)^2
= 0.5 * 2.00 kg * 9.00 m^2/s^2
= 0.5 * 18.00 kg * m^2/s^2
= 9.00 kg * m^2/s^2
Since all the energy is absorbed by the nail, this energy is being converted to an increase in the nail's internal energy and can be equated to the formula:
ΔE = mcΔT
Where:
ΔE = Change in energy
m = Mass of the nail
c = Specific heat capacity of iron
ΔT = Change in temperature
To simplify the calculation, we will assume the specific heat capacity of iron (c) is constant.
Since the nail comes to rest after each swing, we can use the nail's initial kinetic energy as the ΔE value in the energy equation.
ΔE = 9.00 kg * m^2/s^2
Now, we need to calculate the change in temperature (ΔT) of the nail using the equation:
ΔT = ΔE / (mc)
Where:
m = Mass of the nail = 0.100 kg
c = Specific heat capacity of iron
To determine the specific heat capacity of iron, we can use the value of 450 J/kg°C.
Now we can calculate ΔT:
ΔT = 9.00 kg * m^2/s^2 / (0.100 kg * 450 J/kg°C)
Simplifying,
ΔT = (9.00 kg * m^2/s^2) / (0.100 kg * 450 J/kg°C)
ΔT ≈ 2°C
Therefore, the nail's temperature would rise by approximately 2 degrees Celsius after 10 successive swings.