Problem 70 - Master Resource Book in JEE Main Physics

Question

A rectangular plank of mass \(m_{1}\) and height \(a\) is kept on a horizontal surface. Another rectangular plank of mass \(m_{2}\) and height \(b\) is placed over the first plank. The gravitational potential energy of the system is (a) \(\left[m_{1}+m_{2}(a+b)\right] g\) (b) \(\left[\left(\frac{m_{1}+m_{2}}{2} a+m_{2} \frac{b}{2}\right)\right] g\) (c) \(\left[\left(\frac{m_{1}}{2}+m_{2}\right) a+m_{2} \frac{b}{2}\right] g\) (d) \(\left[\left(\frac{m_{1}}{2}+m_{2}\right) a+m_{1} \frac{b}{2}\right] g\)

Step-by-Step Solution

Answer

The answer is (c): \[\left[\left(\frac{m_{1}}{2} + m_{2}\right)a + m_{2}\frac{b}{2}\right]g\].

Step 1: Understanding the Problem

The problem involves two rectangular planks placed on top of each other. We need to calculate the gravitational potential energy of the system.

Step 2: Potential Energy of the First Plank

For an object with mass on a horizontal surface, the center of mass is at half its height. The gravitational potential energy of the first plank, mass \(m_{1}\), and height \(a\), is \(m_{1}g \times \frac{a}{2}\).

Step 3: Potential Energy of the Second Plank

The second plank is placed on top of the first. Its center of mass is at height \(a + \frac{b}{2}\). Thus, its gravitational potential energy is \(m_{2}g \times \left(a + \frac{b}{2}\right)\).

Step 4: Total Gravitational Potential Energy

Add the potential energy of both planks to get the total. This gives us \[\left(m_{1}g \times \frac{a}{2}\right) + \left(m_{2}g \times \left(a + \frac{b}{2}\right)\right)\], simplifying to \[g\left(\frac{m_{1}}{2}a + m_{2}a + m_{2}\frac{b}{2}\right)\].

Step 5: Simplifying the Expression

Combine terms to get a simplified form of the expression. The total expression \[g\left(\frac{m_{1}}{2}a + m_{2}a + m_{2}\frac{b}{2}\right)\] matches option (c): \[\left[\left(\frac{m_{1}}{2} + m_{2}\right)a + m_{2}\frac{b}{2}\right]g\].

Key Concepts

Center of Mass

The center of mass of an object is the point where all of the mass of the object is considered to be concentrated for the purpose of analyzing mechanical properties such as motion and gravitational potential energy. In the context of the exercise, each plank has its own center of mass, which simplifies the calculation of gravitational potential energy. For a uniform rectangular plank, the center of mass is located at its geometric center. Thus:

• For the first plank with height \(a\), the center of mass is at \(\frac{a}{2}\) from its base.
• For the second plank placed on top, its center of mass is at \(a + \frac{b}{2}\), where \(b\) is the height of the second plank.

Understanding this position allows us to correctly calculate how gravity affects each plank's potential energy because it measures the plank's average position relative to the ground surface. By calculating the potential energy from the center of mass, we account for the distribution of mass and height in a straightforward manner.

Rectangular Plank

A rectangular plank in physics is often a simple model used to understand basic mechanical principles due to its uniform shape and properties. In problems involving rectangular planks:

• We assume that the mass is uniformly distributed.
• The gravitational potential energy depends on both its mass and height.
• The plank's dimensions, such as height, play a critical role in determining its center of mass.

In our exercise, the solution exploits these properties by considering each plank separately and calculating the gravitational potential energy based on (a) their mass and (b) the height of their respective centers of mass. This helps illustrate the cumulative effect of stacking objects and serves as a foundational concept in understanding more complex stacking and momentum problems.

Physics Problem Solving

Approaching physics problems systematically is key to finding solutions that are both accurate and insightful. With our exercise involving gravitational potential energy in stacked planks:

• **Identify** the objects involved and their relevant properties – here, the mass and dimensions of the planks.
• **Understand** the concept of gravitational potential energy, which is the energy due to an object's position in a gravitational field.
• **Calculate** individual energies based on the height of each object's center of mass.
• **Combine** the energies to find the total potential energy of the system.

By breaking down each step, the exercise provides a structured approach to solving similar physics problems. This not only aids in understanding the mechanics of the situation but also improves problem-solving skills by following a clear and logical sequence.