When a spring is wound, 100 J of work is done on it but 15 J escapes to the surrounding as heat. What will be the change in internal energy of the spring?
Calculating Internal Energy Change of a Spring
The problem asks for the change in internal energy of a spring when work is done on it and heat escapes to the surroundings. This scenario can be analyzed using the First Law of Thermodynamics.
Understanding the First Law of Thermodynamics
The First Law of Thermodynamics is a statement of the conservation of energy. It relates the change in internal energy ($\Delta U$) of a system to the heat added to the system ($Q$) and the work done by the system ($W$). The mathematical form of the First Law is:
\[ \Delta U = Q - W \]
- $\Delta U$ is the change in the internal energy of the system.
- $Q$ is the heat added to the system. If heat is lost by the system, $Q$ is negative.
- $W$ is the work done by the system. If work is done on the system, $W$ is negative.
Applying the First Law to the Wound Spring
In this specific problem:
- Work is done on the spring. The amount of work done on the spring is 100 J. According to the convention where $W$ is work done by the system, work done on the system is negative. So, the work done by the system is $W = -100$ J.
- Heat escapes to the surroundings. The amount of heat escaping is 15 J. This means heat is lost by the system. According to the convention where $Q$ is heat added to the system, heat lost by the system is negative. So, the heat added to the system is $Q = -15$ J.
Now we can substitute these values into the First Law of Thermodynamics equation:
\[ \Delta U = Q - W \]
\[ \Delta U = (-15 \text{ J}) - (-100 \text{ J}) \]
\[ \Delta U = -15 \text{ J} + 100 \text{ J} \]
\[ \Delta U = +85 \text{ J} \]
Therefore, the change in the internal energy of the spring is +85 J. The internal energy of the spring increases because the work done on it (winding) adds energy, and although some energy is lost as heat, the net change is positive.
Comparing with Options
Let's look at the given options:
- $\Delta U = -115$ J
- $\Delta U = +115$ J
- $\Delta U = +85$ J
- $\Delta U = -85$ J
Our calculated value, $\Delta U = +85$ J, matches the third option.
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