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Chapter 6: Problem 60

At a carnival, you can try to ring a bell by striking a target with a 9.00-kg hammer. In response, a \(0.400-\mathrm{kg}\) metal piece is sent upward toward the bell, which is \(5.00 \mathrm{m}\) above. Suppose that \(25.0 \%\) of the hammer's kinetic energy is used to do the work of sending the metal piece upward. How fast must the hammer be moving when it strikes the target so that the bell just barely rings?

Short Answer

Expert verified
The hammer must move fast enough to deliver 19.62 J of energy to the metal piece.

Step by step solution

01

Understand the Problem

We need to determine how fast the hammer must be moving to transfer sufficient kinetic energy to the metal piece, allowing it to reach a height of 5.00 meters.
02

Calculate the Energy Required

First, we calculate the gravitational potential energy needed for the metal piece to reach 5.00 meters. This is given by the formula:\[ U = mgh \]where \( m = 0.400\, \text{kg} \), \( g = 9.81\, \text{m/s}^2 \), and \( h = 5.00\, \text{m} \). Substituting gives:\[ U = (0.400)(9.81)(5.00) = 19.62\, \text{J} \].

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Key Concepts

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

Potential Energy
Potential energy is the stored energy of an object due to its position and gravitational force acting on it. For the carnival exercise, the metal piece must gain enough potential energy to reach the bell located 5 meters above.

The formula for gravitational potential energy is given by:\[ U = mgh \]where:
  • \( m \) is the mass of the object, here 0.400 kg.
  • \( g \) is the acceleration due to gravity, approximately 9.81 m/s虏.
  • \( h \) is the height to which the object needs to rise, 5.00 meters in this problem.
Substituting the values, we find that the potential energy required is 19.62 joules. This energy is needed to move the metal piece upwards against gravity and reach the bell precisely.
Energy Conservation
Energy conservation is a key principle in physics, stating that energy cannot be created or destroyed, only transformed from one form to another. In the context of our exercise, the kinetic energy of the hammer is partially converted into potential energy to raise the metal piece.

It's given that only 25% of the hammer's kinetic energy transfers to the metal piece. Therefore, we must work backwards to determine how much kinetic energy the hammer initially had.

To find this, we set the useful kinetic energy equal to the potential energy needed:\[ 0.25 imes ext{Kinetic Energy} = 19.62 ext{ J} \]Solving for the total kinetic energy, we get:\[ ext{Kinetic Energy} = \frac{19.62}{0.25} = 78.48 ext{ J} \]Understanding this concept is crucial for solving real-world problems where energy is transformed and shared between systems.
Physics Problems
Solving physics problems often involves identifying the principles at play and applying mathematical formulas. For the carnival bell exercise, we have several steps. First, identify the type of energy transformation occurring when the hammer strikes.

Use the conservation of energy principle to relate the kinetic energy of the hammer to the potential energy required. In such problems, ensure you:
  • Always start by understanding what information is given and what is being asked.
  • Identify the relevant formulas, as we did with the potential energy equation.
  • Apply the principle of energy conservation to transition between forms of energy.
By systematically applying these strategies, you can navigate through complex physics problems and find solutions effectively.

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Most popular questions from this chapter

A gymnast is swinging on a high bar. The distance between his waist and the bar is \(1.1 \mathrm{m},\) as the drawing shows. At the top of the swing his speed is momentarily zero. Ignoring friction and treating the gymnast as if all of his mass is located at his waist, find his speed at the bottom of the swing.

The drawing shows a skateboarder moving at \(5.4 \mathrm{m} / \mathrm{s}\) along a horizontal section of a track that is slanted upward by \(48^{\circ}\) above the horizontal at its end, which is \(0.40 \mathrm{m}\) above the ground. When she leaves the track, she follows the characteristic path of projectile motion. Ignoring friction and air resistance, find the maximum height \(H\) to which she rises above the end of the track.

The hammer throw is a track-and-field event in which a \(7.3 \mathrm{kg}\) ball (the "hammer"), starting from rest, is whirled around in a circle several times and released. It then moves upward on the familiar curving path of projectile motion. In one throw, the hammer is given a speed of \(29 \mathrm{m} / \mathrm{s} .\) For comparison, a .22 caliber bullet has a mass of \(2.6 \mathrm{g}\) and, starting from rest, exits the barrel of a gun at a speed of \(410 \mathrm{m} / \mathrm{s}\). Determine the work done to launch the motion of (a) the hammer and (b) the bullet.

A Water slide is constructed so that swimmers, starting from rest at the top of the slide, leave the end of the slide traveling horizontally. As the drawing shows, one person hits the water \(5.00 \mathrm{m}\) from the end of the slide in a time of 0.500 s after leaving the slide. Ignoring friction and air resistance, find the height \(H\) in the drawing.

A basketball player makes a jump shot. The \(0.600-\mathrm{kg}\) ball is released at a height of \(2.00 \mathrm{m}\) above the floor with a speed of \(7.20 \mathrm{m} / \mathrm{s}\). The ball goes through the net \(3.10 \mathrm{m}\) above the floor at a speed of \(4.20 \mathrm{m} / \mathrm{s} .\) What is the work done on the ball by air resistance, a nonconservative force?
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