Question

The velocity of a small ball of mass \(\mathrm{m}\) and density \(\mathrm{d}_{1}\) when dropped in a container filled with glycerine becomes constants after sometime. What is the viscous force acting on the ball ? (density of glycerine is \(\mathrm{d}_{2}\) ) (A) \(m g\left\\{1-\left(d_{1} / d_{2}\right)\right\\}\) (B) \(m g\left\\{1-\left(d_{2} / d_{1}\right)\right\\}\) (C) \(m g\left\\{d_{1} /\left(d_{1}-d_{2}\right)\right\\}\) (D) \(\operatorname{mg}\left\\{\mathrm{d}_{2} /\left(\mathrm{d}_{1}-\mathrm{d}_{2}\right)\right\\}\)

Short answer

(A) \(mg\\ \{1 - (d_1 / d_2)\\}\)

Step-by-step solution

Identifying the forces acting on the ball

When the ball is moving through the glycerine, it experiences three forces: 1. Gravitational force (weight) = mg, acting downward 2. Buoyant force = the upward force due to the displacement of glycerine, given by \(V \times d_2 \times g\), where V is the volume of the ball 3. Viscous force = the force due to the glycerine acting upward and opposite to the motion of the ball

Finding the terminal velocity

Terminal velocity occurs when the net force acting on the ball becomes zero, and the ball moves with a constant velocity. At terminal velocity, the sum of the buoyant force and the viscous force equals the gravitational force acting on the ball. So, we have: Buoyant force + Viscous force = Gravitational force

Expressing the forces in terms of mass and density

The volume V of the ball can be expressed as \(V = \frac{m}{d_1}\), where m is the mass of the ball and \(d_1\) is its density. Using this, we can rewrite the equation from Step 2 as: \(\frac{m}{d_1} \times d_2 \times g\) + Viscous force = mg

Solving for the viscous force

Rearrange the equation in Step 3 to find the expression for the viscous force: Viscous force = mg - \(\frac{m}{d_1}\) × \(d_2g\) Now, factor out the mg term to simplify: Viscous force = mg\(\\ \{1 - (d_1 / d_2)\\}\) Comparing this expression with the given options, we see that it matches option (A). Therefore, the viscous force acting on the ball is

(A) \(mg\\ \{1 - (d_1 / d_2)\\}\)

Key concepts

Viscous Force

When a small ball is dropped into a fluid like glycerine, it encounters a force that opposes its motion, known as the viscous force. This happens due to the internal friction between layers of the glycerine and the ball moving through it.
Understanding viscous force helps to explain why liquids of different thickness (viscosity) affect moving objects differently.

• In the context of the problem, the ball experiences this force upwards.
• The formula derived for the viscous force is: \[ ext{Viscous Force} = mg \{ 1 - \frac{d_1}{d_2} \} \]

This formula shows that the viscous force depends on the mass of the ball, the gravitational acceleration, and the densities of the ball and the fluid. The term \( \frac{d_1}{d_2} \) indicates the ratio of the ball's density to that of the glycerine fluid.

Buoyant Force

The buoyant force is an upward force exerted by a fluid, opposing the weight of an object in the fluid. It occurs because the pressure below the object is greater than above, creating a net upward push.
To determine the buoyant force acting on the ball, we use Archimedes' principle:

• Buoyant Force = Volume of the ball \( \times \text{Density of fluid} \times g\)


• For our scenario, the formula becomes \( \frac{m}{d_1} \times d_2 \times g \)

Here, \( V = \frac{m}{d_1} \) represents the volume of the ball. Thus, the buoyant force relies on both the fluid's density and the ball's volume. Understanding buoyant forces explains why objects float or sink depending on their density relative to the fluid.

Density

Density is a measure of mass per unit volume. It's symbolized by \( \,d\), and is crucial in determining how substances behave in fluids.
In our exercise, the ball's density \( d_1 \) is compared with the fluid's density \( d_2 \).

• An object will float if its density is less than the fluid's, and sink if it is more.
• The concept of density helps determine both the buoyant and viscous forces.
• The ratio \( \frac{d_1}{d_2} \) plays a key role in our formula for viscous force.

For example, a high-density object will likely experience more gravitational pull than a low-density fluid, thus potentially sinking. This comparison illustrates why density is an essential factor in physics when analyzing floating and sinking behavior.

Net Force

Net force is the overall force acting on an object, and it's a vital concept in understanding motion. It is the vector sum of all the individual forces.
For the ball in glycerine, the forces active are:

• Gravitational Force acting downwards.
• Buoyant Force and Viscous Force acting upwards.
• Net Force = Gravitational Force - (Buoyant Force + Viscous Force).

At terminal velocity, when the object falls at constant speed, the net force becomes zero. This equality is used to solve for the viscous force, as it ensures that the upward forces exactly balance the downward gravitational force. By understanding net force, it becomes clearer how and when an object will accelerate or move at a steady pace.