Problem 2 - Classical Mechanics

Question

Water in a rotating container of radius \(50 \mathrm{~mm}\) is \(30 \mathrm{~mm}\) lower in the centre than at the edge. Find the angular velocity of the container.

Step-by-Step Solution

Answer

Answer: The angular velocity of the container is approximately 4.85 rad/s.

Step 1: Forces acting on the water

The forces acting on the water in the container are the gravitational force and the centripetal force. The gravitational force acts vertically downwards, while the centripetal force acts horizontally towards the center of the container as it rotates. The centripetal force is given by the equation \(F_{c} = m \omega^2 r\) where \(F_{c}\) is the centripetal force, \(m\) is the mass of the water, \(\omega\) is the angular velocity, and \(r\) is the distance from the center of the container.

Step 2: Establish the relationship between angular velocity and the shape of the water

Due to the rotational motion of the container, the water's shape is determined by an equilibrium between centrifugal force and gravitational force. Let \(h\) be the difference between the height at the edge and the height in the center, then we can relate gravitational force with centrifugal force as follows: \(\rho g h = \rho (\omega r)^2/2\) Where \(\rho\) is the water density, \(g\) is the acceleration due to gravity (approximately \(9.8 \mathrm{ms^{-2}}\)), \(h\) is the height difference, which is \(30 \mathrm{mm}\), and \(r\) is the radius of the container, which is \(50 \mathrm{mm}\). Note that we've divided by 2 on the right side of the equation because we're calculating the difference in heights at the center and the edge.

Step 3: Calculate the angular velocity

Now, we can solve the equation for angular velocity: \(\omega^2 = \frac{2gh}{r^2}\) Substituting the known values, we have: \(\omega^2 = \frac{2(9.8 \mathrm{ms^{-2}})(0.03 \mathrm{m})}{(0.05 \mathrm{m})^2}\) Solving for \(\omega\): \(\omega = \sqrt{\frac{2(9.8)(0.03)}{(0.05)^2}} = \sqrt{23.52} \approx 4.85 \mathrm{rad/s}\) Therefore, the angular velocity of the container is approximately \(4.85 \mathrm{rad/s}\).

Key Concepts

Centripetal Force

When we talk about rotational motion, the term "centripetal force" often comes up. Imagine swinging a ball on a string in a circular path. The string exerts an inward force on the ball, keeping it moving in a circle. This is similar to what happens in the rotating water container.Centripetal force is essential for any object moving in a circle. It acts toward the center of the circle, which means it constantly changes the direction of the moving object, even though its speed might remain constant. To find the centripetal force, the equation used is:

• \(F_c = m \omega^2 r\)
• Where \(F_c\) is the centripetal force, \(m\) is mass, \(\omega\) is angular velocity, and \(r\) is the radius of the circular path.

Understanding this force helps explain why the water in the rotating container rises at the edge—it’s the centripetal force that pushes it outward.

Centrifugal Force

Centrifugal force is often described as the "opposite" of centripetal force. Imagine you’re in a car taking a sharp turn; you feel like you’re pushed outward. That's centrifugal force. Though it isn’t a real force like gravity or electromagnetism, it's a perceived effect that arises in a rotating system. In the case of the water container, while centripetal force pulls the water inward, centrifugal "force" is why the water tends to move outward against the walls:

• The water experiences this outward pull due to its inertia, as it attempts to move in a straight line.
• It's an apparent force experienced in the rotating reference frame of the container.

This apparent force causes the water’s surface to shape like a parabola, with the center lower than the edges.

Rotational Motion

Rotational motion occurs when an object spins around an axis. It’s all around us, from the Earth spinning on its axis to the wheels on a car. In the case of the rotating water container, the rotational motion causes the water to redistribute itself unevenly.Key properties of rotational motion include:

• Angular velocity \(\omega\), which indicates how fast an object rotates.
• Radius \(r\), the distance from the axis of rotation.

The angular velocity determines how quickly the circular path is completed, impacting how the forces act in the system. In the water container example, a higher angular velocity means a greater centripetal force, causing the water to rise more sharply at the edges.

Gravitational Force

Gravitational force is a fundamental force that acts between two masses, pulling them toward one another. We feel this as weight here on Earth. In the context of the rotating container, gravity is ever-present, always acting downward on the water.Gravity interacts with the other forces, playing a crucial role:

• It competes with the centrifugal "force", trying to keep the water level.
• The equation \(\rho g h = \rho (\omega r)^2/2\) describes this balance in the rotating container system.

Understanding gravity's role helps us comprehend why the water's surface in the rotating container isn’t only determined by centrifugal effects—it’s a delicate balance between the upward and downward forces.