A long metal rod of length l and relative density (s) is held vertically with its lower end touching the surface of water. The speed of the rod when it sinks in the water is ? Answer:(2gl(1-1/2s))^1/2

To determine the speed of a long metal rod as it sinks in water, we need to consider several factors, including the rod's density, the buoyant force acting on it, and the principles of motion. Let's break this down step by step to understand how we arrive at the formula for the speed of the rod.

Understanding the Forces at Play

When the rod is placed in water, two primary forces act on it:

• Gravitational Force (Weight): This is the force due to gravity acting on the rod, calculated as \( W = mg \), where \( m \) is the mass of the rod and \( g \) is the acceleration due to gravity.
• Buoyant Force: According to Archimedes' principle, the buoyant force is equal to the weight of the water displaced by the submerged part of the rod. This can be expressed as \( F_b = \rho_{water} \cdot V_{displaced} \cdot g \), where \( \rho_{water} \) is the density of water and \( V_{displaced} \) is the volume of the rod submerged in water.

Calculating the Forces

Let's denote the length of the rod as \( l \) and its relative density as \( s \). The density of the rod can be expressed as \( \rho_{rod} = s \cdot \rho_{water} \). The volume of the rod is \( V_{rod} = A \cdot l \), where \( A \) is the cross-sectional area of the rod. When the rod is fully submerged, the volume of water displaced is equal to the volume of the rod.

The weight of the rod is:

\( W = \rho_{rod} \cdot V_{rod} \cdot g = s \cdot \rho_{water} \cdot (A \cdot l) \cdot g \)

The buoyant force acting on the rod when it is submerged is:

\( F_b = \rho_{water} \cdot V_{rod} \cdot g = \rho_{water} \cdot (A \cdot l) \cdot g \)

Setting Up the Equation of Motion

When the rod is released, it will accelerate downwards until it reaches a terminal velocity, where the net force acting on it becomes zero. The net force can be expressed as:

\( F_{net} = W - F_b \)

Substituting the expressions for weight and buoyant force, we have:

\( F_{net} = (s \cdot \rho_{water} \cdot A \cdot l \cdot g) - (\rho_{water} \cdot A \cdot l \cdot g) \)

Factoring out common terms gives:

\( F_{net} = \rho_{water} \cdot A \cdot l \cdot g (s - 1) \)

Applying Newton's Second Law

According to Newton's second law, the net force is also equal to mass times acceleration:

\( F_{net} = m \cdot a \)

Setting these two expressions for net force equal to each other, we have:

\( m \cdot a = \rho_{water} \cdot A \cdot l \cdot g (s - 1) \)

Now, the mass of the rod can be expressed as:

\( m = s \cdot \rho_{water} \cdot A \cdot l \)

Finding the Speed of the Rod

Substituting this expression for mass back into our equation gives:

\( s \cdot \rho_{water} \cdot A \cdot l \cdot a = \rho_{water} \cdot A \cdot l \cdot g (s - 1) \)

We can simplify this by canceling out common terms:

\( s \cdot a = g (s - 1) \)

Solving for acceleration \( a \) yields:

\( a = \frac{g (s - 1)}{s} \)

Integrating to Find Speed

To find the speed of the rod as it sinks, we can integrate the acceleration over time. Assuming the rod starts from rest, the speed \( v \) can be expressed as:

\( v = \sqrt{2 \cdot a \cdot d} \)

Where \( d \) is the distance fallen. Substituting our expression for acceleration gives:

\( v = \sqrt{2 \cdot \frac{g (s - 1)}{s} \cdot l} \)

After simplifying, we arrive at the final expression for the speed of the rod as it sinks in water:

\( v = \sqrt{2gl(1 - \frac{1}{2}s)} \)

This formula encapsulates the relationship between the rod's density, the gravitational force, and the distance it travels as it sinks in water. Understanding these principles not only helps in solving this problem but also lays the groundwork for exploring fluid dynamics and buoyancy further.