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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 small car with mass 0.800 kg travels at constant speed on the inside of a track that is a vertical circle with radius 5.00 m (Fig. E5.45). If the normal force exerted by the track on the car when it is at the top of the track (point \(B\)) is 6.00 N, what is the normal force on the car when it is at the bottom of the track (point \(A\))?

A box of bananas weighing 40.0 N rests on a horizontal surface. The coefficient of static friction between the box and the surface is 0.40, and the coefficient of kinetic friction is 0.20. (a) If no horizontal force is applied to the box and the box is at rest, how large is the friction force exerted on it? (b) What is the magnitude of the friction force if a monkey applies a horizontal force of 6.0 N to the box and the box is initially at rest? (c) What minimum horizontal force must the monkey apply to start the box in motion? (d) What minimum horizontal force must the monkey apply to keep the box moving at constant velocity once it has been started? (e) If the monkey applies a horizontal force of 18.0 N, what is the magnitude of the friction force and what is the box's acceleration?

An airplane flies in a loop (a circular path in a vertical plane) of radius 150 m. The pilot's head always points toward the center of the loop. The speed of the airplane is not constant; the airplane goes slowest at the top of the loop and fastest at the bottom. (a) What is the speed of the airplane at the top of the loop, where the pilot feels weightless? (b) What is the apparent weight of the pilot at the bottom of the loop, where the speed of the airplane is 280 km/h? His true weight is 700 N.

An astronaut is inside a 2.25 \(\times\) 10\(^6\) kg rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound (331 m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than 4\(g\). (a) What is the maximum initial thrust this rocket's engines can have but just barely avoid blackout? Start with a free-body diagram of the rocket. (b) What force, in terms of the astronaut's weight \(w\), does the rocket exert on her? Start with a free-body diagram of the astronaut. (c) What is the shortest time it can take the rocket to reach the speed of sound?

One problem for humans living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates "artificial gravity" at the outside rim of the station. (a) If the diameter of the space station is 800 m, how many revolutions per minute are needed for the "artificial gravity" acceleration to be 9.80 m/s\(^2\)? (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface (3.70 m/s\(^2\)). How many revolutions per minute are needed in this case?

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