A 500.0 kg object is attached by a rope through a pulley to a paddle-wheel shaft that is placed in a well-insulated tank holding 25.0 kg of water. The object is allowed to fall, causing the paddle wheel to rotate, churning the water. If the object falls a vertical distance of 100.0 m at constant speed, what is the temperature change of the water?

4.7 C°

To determine the temperature change of the water, we need to use the concept of potential energy and convert it into thermal energy.

Step 1: Calculate the potential energy of the object
The potential energy of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Given:
Mass of the object (m) = 500.0 kg
Height (h) = 100.0 m
Acceleration due to gravity (g) = 9.8 m/s^2
Substituting the values into the formula, we have:
PE = (500.0 kg) * (9.8 m/s^2) * (100.0 m)
PE = 490,000 J

Step 2: Calculate the amount of work done on the water
Since the potential energy is transferred to the paddle-wheel and then to the water in the tank, we can calculate the work done on the water.
The work done (W) is given by the formula W = PE = Δthermal energy.
Note: The thermal energy gained by the water is equal to the work done on it.
Thus, W = PE = 490,000 J.

Step 3: Calculate the change in temperature of the water
The amount of thermal energy gained by the water can be calculated using the specific heat capacity formula Q = mcΔT, where Q is the thermal energy gained, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature.

Given:
Mass of the water (m) = 25.0 kg
Specific heat capacity of water (c) = 4,186 J/kg·°C (approximately)
ΔT = ?

Rearranging the formula, ΔT = Q / (mc).

Substituting the values, we have:
ΔT = 490,000 J / [(25.0 kg) * (4,186 J/kg·°C)]
ΔT ≈ 4.7°C

Therefore, the temperature of the water increases by approximately 4.7°C.

To determine the temperature change of the water, we need to calculate the amount of potential energy converted into thermal energy.

First, let's find the potential energy (PE) of the falling object using the formula:

PE = m * g * h

Where:
m = mass of the object = 500.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height or vertical distance fallen = 100.0 m

PE = 500.0 kg * 9.8 m/s^2 * 100.0 m
PE = 490,000 J (joules)

Since the object is falling at a constant speed, all of its potential energy is being converted into thermal energy in the water.

Next, we need to find the specific heat capacity of water, which is the amount of heat energy required to raise the temperature of a given mass of water by a certain amount. The specific heat capacity of water is approximately 4,184 J/(kg°C).

Now, we can calculate the temperature change (ΔT) using the formula:

ΔT = Q / (m_water * c_water)

Where:
Q = heat energy transferred to the water (in joules)
m_water = mass of the water = 25.0 kg
c_water = specific heat capacity of water = 4,184 J/(kg°C)

Since all the potential energy is converted into thermal energy, Q is equal to the potential energy (PE) of the falling object.

ΔT = 490,000 J / (25.0 kg * 4,184 J/(kg°C))
ΔT ≈ 11.7 °C

Therefore, the temperature change of the water is approximately 11.7 °C.