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You tie a cord to a pail of water and swing the pail in a vertical circle of radius 0.600 m. What minimum speed must you give the pail at the highest point of the circle to avoid spilling water?

Short Answer

Expert verified
2.43 m/s

Step by step solution

01

Understanding the Problem

To avoid spilling, the centrifugal force must be equal to or greater than the gravitational force at the top of the circle. Thus, the minimum speed is such that the gravitational force is exactly balanced by the required centripetal force to maintain circular motion.
02

Applying Formulas

At the top of the circle, the required centripetal force is provided by the gravitational force. The gravitational force is calculated by \( F_g = mg \) and the centripetal force by \( F_c = \frac{mv^2}{r} \). To avoid spilling, set these equal: \( mg = \frac{mv^2}{r} \).
03

Simplifying the Equation

We can cancel out the mass \( m \) from both sides of the equation, leaving \( g = \frac{v^2}{r} \). This simplifies finding the minimum speed \( v \) the pail needs at the top to avoid spillage.
04

Solving for Minimum Speed

Rearrange the equation \( g = \frac{v^2}{r} \) to solve for \( v \): \( v^2 = gr \). Therefore, \( v = \sqrt{gr} \). Given \( r = 0.600 \) m and using \( g = 9.81 \) m/s², substitute these into the equation: \( v = \sqrt{9.81 \times 0.600} \).
05

Calculating the Result

Compute \( v = \sqrt{5.886} \) which gives \( v \approx 2.43 \) m/s. This is the minimum speed required at the top of the circle to avoid spilling water from the pail.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Force
Gravitational force plays a crucial role in all kinds of motion, especially when objects move in a circle. This force is the invisible attraction between two objects with mass.
The more massive an object, the greater the gravitational pull it exerts on another object.
  • The force of gravity acts downward towards the center of the Earth.
  • It is calculated using the formula: \( F_g = mg \), where \( m \) is mass and \( g \) is the acceleration due to gravity.
  • The standard value of \( g \) on Earth is approximately \( 9.81 \text{ m/s}^2 \).
For an object in vertical circular motion, like our pail of water, gravitational force is especially important at the top of the circle, providing the necessary force to keep it in motion.
In essence, at the highest point of the swing, gravity aids in providing the centripetal force required to maintain the pail's circular path.
Circular Motion
Circular motion is when an object travels along a curved path or a circle. In this kind of motion, a centripetal force is necessary to keep the object moving in a circle, constantly pulling it towards the center.
Examples include a satellite orbiting a planet or a car turning around a curved road.
  • The force keeping the object in the circular path is called centripetal force.
  • This force is directed towards the center of the circle and is responsible for changing the direction of the object's velocity without altering its speed.
  • For a pail of water swung in a vertical circle, the centripetal force at the top of the circle is provided by gravity.
  • Centripetal force can be calculated using the formula: \( F_c = \frac{mv^2}{r} \).
Circular motion isn't only confined to gravity. Friction or tension can also act as centripetal forces in other scenarios, like driving or swinging.
Minimum Speed Calculation
Determining the minimum speed in circular motion scenarios is necessary to keep an object moving along its path without faltering. In this exercise, our goal is to ensure the pail at the highest point does not spill the water.
This is achieved by having the gravitational force effectively turn into the centripetal force needed to maintain the motion.
  • The key equation to determine minimum speed in this context is derived by setting gravitational force equal to centripetal force: \( mg = \frac{mv^2}{r} \).
  • From this equation, you can simplify to find the speed: \( v = \sqrt{gr} \).
  • This formula ensures that the speed is enough to counteract the gravitational pull that tries to spill the water.
  • Substituting \( g = 9.81 \text{ m/s}^2 \) and \( r = 0.600 \text{ m} \) gives a minimum speed of approximately 2.43 m/s.
This calculation is important in various areas. For instance, roller coasters must reach minimum speeds at certain points to keep riders safely looped, and vehicles on curved roads require adequate speeds to prevent skidding.

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Most popular questions from this chapter

A physics major is working to pay her college tuition by performing in a traveling carnival. She rides a motorcycle inside a hollow, transparent plastic sphere. After gaining sufficient speed, she travels in a vertical circle with radius 13.0 m. She has mass 70.0 kg, and her motorcycle has mass 40.0 kg. (a) What minimum speed must she have at the top of the circle for the motorcycle tires to remain in contact with the sphere? (b) At the bottom of the circle, her speed is twice the value calculated in part (a). What is the magnitude of the normal force exerted on the motorcycle by the sphere at this point?

A 550-N physics student stands on a bathroom scale in an elevator that is supported by a cable. The combined mass of student plus elevator is 850 kg. As the elevator starts moving, the scale reads 450 N. (a) Find the acceleration of the elevator (magnitude and direction). (b) What is the acceleration if the scale reads 670 N? (c) If the scale reads zero, should the student worry? Explain. (d) What is the tension in the cable in parts (a) and (c)?

An 8.00-kg block of ice, released from rest at the top of a 1.50-m-long frictionless ramp, slides downhill, reaching a speed of 2.50 m/s at the bottom. (a) What is the angle between the ramp and the horizontal? (b) What would be the speed of the ice at the bottom if the motion were opposed by a constant friction force of 10.0 N parallel to the surface of the ramp?

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