Problem 14 - College Physics

Question

Stay dry! You tie a cord to a pail of water, and you swing the pail in a vertical circle of radius 0.600 \(\mathrm{m} .\) What minimum speed must you give the pail at the highest point of the circle if no water is to spill from it? Start with a free-body diagram of the water at its highest point.

Step-by-Step Solution

Answer

Minimum speed is 2.43 m/s.

Step 1: Understand the Problem

We need to find the minimum speed to swing the pail of water in a vertical circle without spilling it at the highest point. At this point, the only forces acting on the water are gravity and the centripetal force. This requires analyzing the forces when the pail is at the top of the circle.

Step 2: Draw the Free-Body Diagram

At the highest point of the circle, the forces acting on the water are the gravitational force (weight) downward, and the centripetal force required for circular motion, which also points downward. The gravitational force must provide the necessary centripetal force to keep the water in the pail.

Step 3: Set Up the Equation for Forces

The gravitational force must be equal to or greater than the centripetal force required to keep the pail moving in a circle. The equation is: \[ mg = rac{mv^2}{r} \]where \( m \) is the mass of the water, \( g \) is the acceleration due to gravity \( 9.8 \, \mathrm{m/s^2} \), \( v \) is the velocity, and \( r \) is the radius of the circle, 0.600 m.

Step 4: Solve for the Minimum Speed

Rearrange the equation to solve for the velocity \( v \):\[ mg = rac{mv^2}{r} \]Cancel the mass \( m \) from both sides:\[ g = rac{v^2}{r} \]Solve for \( v \):\[ v^2 = gr \]\[ v = \sqrt{gr} \]Substitute \( g = 9.8 \, \mathrm{m/s^2} \) and \( r = 0.600 \, \mathrm{m} \):\[ v = \sqrt{9.8 \, \mathrm{m/s^2} \times 0.600 \, \mathrm{m}} \]\[ v = \sqrt{5.88} \]\[ v \approx 2.43 \, \mathrm{m/s} \]

Step 5: Conclusion

The minimum speed required at the highest point of the circle to ensure no water spills from the pail is approximately 2.43 m/s.

Key Concepts

Centripetal Force

When you swing a pail in a circle, centripetal force is a key component that keeps the pail moving in its circular path. Centripetal force is the attraction needed to make something move in a circle rather than move off in a straight line. It acts inward, towards the center of the circle. This force is required to change the direction of the object's velocity as it moves along a curved path, preventing it from flying outward.

In the case of the swinging pail, the centripetal force is essential at the highest point of the circle. At this point, the gravitational force helps provide the necessary centripetal force. Without enough centripetal force, the water would not stay inside the pail. If you're swinging the pail too slowly, gravity wins out, and the water will spill. Hence, understanding centripetal force helps ensure the right speed while swinging an object in a circular motion.

To calculate the centripetal force, we often use the equation:

• \( F_c = \frac{mv^2}{r} \)

where:

• \( F_c \) is the centripetal force,
• \( m \) is the mass of the object,
• \( v \) is the velocity of the object, and
• \( r \) is the radius of the circle.

This formula helps determine the balance of forces needed to keep objects moving in a circle.

Gravitational Force

Gravitational force is the natural phenomenon by which all things with mass are brought towards one another, including objects in space. In simpler terms, it is the force that pulls objects toward the Earth's center. This force is a crucial player in many scenarios, including circular motion.

In the example of swinging a pail, gravitational force acts downwards on both the pail and the water within it. At the highest point in a vertical circle, gravity is not merely trying to pull the water out of the pail. Instead, it also acts as part of the centripetal force necessary to keep the water in its circular path. This dual role is vital for understanding why the water stays in the pail when swung above the head.

The force of gravity acting on an object is described by the equation:

• \( F_g = mg \)

where:

• \( F_g \) is the gravitational force,
• \( m \) is the mass of the object, and
• \( g \) is the acceleration due to gravity, approximately \( 9.8 \, \mathrm{m/s^2} \) on Earth.

This equation is fundamental in calculating how different forces interact during circular motion.

Free-Body Diagram

A free-body diagram is an essential tool in physics that helps visualize the forces acting on a single object without considering the forces the object exerts on other things. It simplifies complex problems into manageable representations, which are crucial for solving physics problems involving forces.

In our example, the free-body diagram of the pail of water at its highest point shows two main forces:

• Gravitational Force (downward)
• Centripetal Force (also downward)

By illustrating these forces, a free-body diagram helps students understand the dynamic balance at the highest point of a circular path. Each arrow in this diagram represents the direction and relative magnitude of these forces.

Using a free-body diagram can make it simpler to write equations that equate forces to solve for unknowns, such as the minimum speed needed for the swinging motion. With practice, constructing these diagrams becomes easier and is a valuable skill for tackling physics problems.