Problem 17 - Master Resource Book in JEE Main Physics

Question

A \(5 \mathrm{~kg}\) stone of relative density 3 is resting at the bed of a lake. It is lifted through a height of \(5 \mathrm{~m}\) in the lake. If \(g=10 \mathrm{~ms}^{-2}\), then the work done is (a) \(\frac{500}{3} \mathrm{~J}\) (b) \(\frac{350}{3} \mathrm{~J}\) (c) \(\frac{750}{3} \mathrm{~J}\) (d) zero

Step-by-Step Solution

Answer

The work done is \(\frac{500}{3} \text{ J}\).

Step 1: Understanding Relative Density

Relative density (or specific gravity) is the ratio of the density of a substance to the density of a reference substance. Here, the relative density of the stone is given as 3, meaning its density is 3 times that of water. Since the density of water is approximately \(1000 \text{ kg/m}^3\), the density of the stone is \(3000 \text{ kg/m}^3\).

Step 2: Calculate the Volume of the Stone

Using the formula for mass \((\text{mass} = \text{density} \times \text{volume})\), we find the volume \(V\) of the stone. Given the mass \(m\) as \(5 \text{ kg}\), we express: \[ V = \frac{m}{\text{density}} = \frac{5 \text{ kg}}{3000 \text{ kg/m}^3} = \frac{1}{600} \text{ m}^3. \]

Step 3: Calculate the Buoyant Force

The buoyant force is given by Archimedes' principle as the weight of the displaced water. Since the volume of the stone is \(\frac{1}{600} \text{ m}^3\), the buoyant force \(F_b\) is: \[ F_b = \text{density of water} \times V \times g = 1000 \text{ kg/m}^3 \times \frac{1}{600} \text{ m}^3 \times 10 \text{ m/s}^2 = \frac{50}{3} \text{ N}. \]

Step 4: Calculate the Weight of the Stone

The weight \(W_s\) of the stone is the mass times gravity: \[ W_s = mg = 5 \text{ kg} \times 10 \text{ m/s}^2 = 50 \text{ N}. \]

Step 5: Determine the Net Force Required to Lift the Stone

The net force \(F_n\) required to lift the stone is the difference between its weight and the buoyant force: \[ F_n = W_s - F_b = 50 \text{ N} - \frac{50}{3} \text{ N}. \] Calculating further, \[ F_n = \frac{150}{3} \text{ N} - \frac{50}{3} \text{ N} = \frac{100}{3} \text{ N}. \]

Step 6: Compute the Work Done to Lift the Stone

Work done \(W\) is calculated as the product of the net force and the height the stone is lifted: \[ W = F_n \times h = \frac{100}{3} \text{ N} \times 5 \text{ m} = \frac{500}{3} \text{ J}. \]

Key Concepts

Buoyant Force

Buoyant force is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is what makes objects like boats float on water. It is crucial in determining how easy or difficult it will be to lift an object from the bottom of a lake or any fluid. If you've ever tried to lift something underwater, you might have noticed that it seems lighter. That's because the buoyant force is helping you. The force occurs because a fluid pushes up on any object placed in it. The amount of fluid pushed out of the way—or displaced—corresponds to the volume of the object submerged in the fluid. The magnitude of the buoyant force can be calculated using the formula: - \[ F_b = \text{Density of fluid} \times \text{Volume of object} \times g \] - In the problem involving the stone lifted in a lake, the buoyant force represents the weight of the water displaced by the stone. In essence, it's this upward force that reduces the effective weight of the stone in the water.

Archimedes' Principle

Archimedes' principle is a fundamental concept in fluid mechanics. It states that any object completely or partially submerged in a fluid is buoyed up by a force equal to the weight of the fluid that the object displaces. This principle can be easily observed in our everyday lives: think about how you float easier in the sea (with its dense saltwater) than in a swimming pool. This is because the sea water pushes up more due to its density, compared to pool water. - For example: - When a stone is submerged in water, the upward force it experiences (buoyant force) equals the weight of the volume of water the stone displaces. - This principle helps in calculating whether an object will float or sink and how much of it will be underwater. Archimedes' principle is essential for engineers when designing ships and underwater structures. Understanding this concept allows us to predict and calculate buoyancy, ensuring structures remain stable and afloat.

Work Done

Work done on an object is the product of the net force acting on it and the distance over which the force acts. In physics, work is done when a force causes an object to move. It is calculated using the formula: - \[ W = F \times d \] - where \( W \) is the work done, \( F \) is the force applied, and \( d \) is the distance moved by the object in the direction of the force. In the problem of lifting a stone from a lake bed, the work done involves lifting the stone a vertical distance through the water. The net force required to lift the stone is influenced by both its weight and the buoyant force. Therefore, the work done is calculated by multiplying this net force by the height the stone is lifted. Key points to remember:

• Work is done only when the object moves in the direction of the applied force.
• The lift would require less work due to the buoyant force reducing the effective weight of the stone.
• Work can be measured in joules in the metric system, reflecting the energy required to move the object.

Understanding the concept of work done helps us measure energy changes in various physical processes, emphasizing the relationship between force and motion.