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Chapter 7: Problem 49

The Great Sandini is a \(60 \mathrm{~kg}\) circus performer who is shot from a cannon (actually a spring gun). You don't find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of \(1100 \mathrm{~N} / \mathrm{m}\) that he will compress with a force of \(4400 \mathrm{~N}\). The inside of the gun barrel is coated with Teflon, so the average friction force will be only \(40 \mathrm{~N}\) during the \(4.0 \mathrm{~m}\) he moves in the barrel. At what speed will he emerge from the end of the barrel, \(2.5 \mathrm{~m}\) above his initial rest position?

Short Answer

Expert verified
The performer will emerge from the barrel at a speed of about 16.1 m/s.

Step by step solution

01

Calculate the spring compression distance

Initially, we calculate the distance the spring is compressed by using the formula for the force constant of a spring which is \( F = k \cdot x \), where \( F = 4400 \mathrm{~N} \), \( k = 1100 \mathrm{~N/m} \) and \( x \) is the compression. Solving for \( x \), we get \( x = F / k = 4400~\mathrm{N} / 1100~\mathrm{N/m} = 4.0~\mathrm{m} \).
02

Calculate the spring potential energy

Calculate the potential energy stored in the spring at the point of maximum compression. This is given by the formula \( PE_{spring} = 0.5 \cdot k \cdot x^2 \), where \( k = 1100 \mathrm{~N/m} \) and \( x = 4.0 \mathrm{~m} \). Thus, \( PE_{spring} = 0.5 \cdot 1100 \mathrm{~N/m} \cdot (4.0 \mathrm{~m})^2 = 8800~\mathrm{J} \).
03

Calculate the work done against friction

Calculate the work done against friction as the performer moves through the barrel. This is given by the formula \( W_{friction} = F_{friction} \cdot d \), where \( F_{friction} = 40 \mathrm{~N} \) and \( d = 4.0 \mathrm{~m} \). So, \( W_{friction} = 40 \mathrm{~N} \cdot 4.0 \mathrm{~m} = 160 \mathrm{~J} \).
04

Calculate the gravitational potential energy

Calculate the gravitational potential energy at the point the performer emerges from the barrel. This is given by \( PE_{gravity} = m \cdot g \cdot h \), where \( m = 60 \mathrm{~kg} \), \( g = 9.8 \mathrm{~m/s^2} \) and \( h = 2.5 \mathrm{~m} \). Thus, \( PE_{gravity} = 60 \mathrm{~kg} \cdot 9.8 \mathrm{~m/s^2} \cdot 2.5 \mathrm{~m} = 1470~\mathrm{J} \).
05

Use Principle of Conservation of Mechanical Energy

The performer's initial total energy is equal to the potential energy of the spring, which is then transformed into kinetic energy, friction and gravitational potential energy as he emerges from the barrel. So we have \( PE_{spring} = KE + W_{friction} + PE_{gravity} \). Rearranging the equation for kinetic energy we get, \( KE = PE_{spring} - W_{friction} - PE_{gravity} = 8800~\mathrm{J} - 160~\mathrm{J} - 1470~\mathrm{J} = 7170~\mathrm{J} \).
06

Finally compute the speed

The performer's final speed can be calculated using the formula for kinetic energy: \( KE = 0.5 \cdot m \cdot v^2 \) where, \( m = 60~\mathrm{kg} \) and \( KE = 7170~\mathrm{J} \). Solving for speed \( v \), we get \( v = \sqrt{(2 \cdot KE) / m} = \sqrt{(2 \cdot 7170~\mathrm{J}) / 60~\mathrm{kg}} = 16.1~\mathrm{m/s} \). Hence, the performer will emerge from the barrel at a speed of about 16.1 m/s.

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Key Concepts

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

Spring Potential Energy
Spring potential energy is the energy stored in a spring that has been compressed or stretched. In our example, the Great Sandini's cannon uses a large spring, which is compressed by his force. The more the spring is compressed, the more energy is stored in it. This energy is then released to shoot Sandini out of the cannon.

The mathematical expression for spring potential energy is \[ PE_{spring} = 0.5 \cdot k \cdot x^2 \] where
  • \( k \) is the spring constant (a measure of the spring's stiffness), and
  • \( x \) is the displacement or compression from the spring's original position.
In this calculation, the energy stored is converted into other forms of energy, making the concept of spring potential energy crucial for understanding Sandini's flight dynamics.
Friction Force
Friction force is a force that opposes the motion of objects sliding against each other. While striving for a high-speed launch, it is essential to consider the influence of friction. Friction within Sandini's launch setup is minimized by coating the barrel in Teflon.

Even though minimized, friction doesn't disappear. The work done against friction is calculated as: \[ W_{friction} = F_{friction} \cdot d \] where
  • \( F_{friction} \) is the force due to friction, and
  • \( d \) is the distance Sandini travels in the barrel.
This calculation indicates energy loss due to friction, which reduces the kinetic energy available for launching Sandini. Understanding this concept helps in optimizing the design to ensure a successful performance.
Gravitational Potential Energy
Gravitational Potential Energy (GPE) is the energy an object possesses due to its height above the ground. As Sandini exits the cannon, he reaches a height of 2.5 meters, which adds to the potential energy in this system.

The formula for calculating gravitational potential energy is: \[ PE_{gravity} = m \cdot g \cdot h \] where:
  • \( m \) is the mass of Sandini,
  • \( g \) is the acceleration due to gravity (approximately \(9.8\ \mathrm{m/s^2}\)), and
  • \( h \) is the height gained.
Recognizing gravitational potential energy helps assess the effect of height on the energy budget. It decreases the kinetic energy because some of the spring potential energy is converted into GPE as Sandini ascends.
Kinetic Energy
Kinetic energy is the energy of motion. Upon leaving the cannon, Sandini's movement is driven by the kinetic energy obtained from the initially stored spring potential energy. The speed at which he emerges is a direct indicator of his kinetic energy.

The kinetic energy of an object is calculated using the formula: \[ KE = 0.5 \cdot m \cdot v^2 \] where
  • \( m \) is the mass, and
  • \( v \) is the velocity of the object.
Through the principle of energy conservation, we deduce that the total energy at the beginning (spring potential energy) minus the energy losses (friction and GPE) equals the kinetic energy. This principle allows us to determine Sandini's speed as he leaves the cannon, enabling precise performance planning.

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Most popular questions from this chapter

The maximum height a typical human can jump from a crouched start is about 60 cm. By how much does the gravitational potential energy increase for a 72 kg person in such a jump? Where does this energy come from?

CALC The potential energy of two atoms in a diatomic molecule is approximated by \(U(r)=\left(a / r^{12}\right)-\left(b / r^{6}\right),\) where \(r\) is the spacing between atoms and \(a\) and \(b\) are positive constants. (a) Find the force \(F(r)\) on one atom as a function of \(r .\) Draw two graphs: one of \(U(r)\) versus \(r\) and one of \(F(r)\) versus \(r\). (b) Find the equilibrium distance between the two atoms. Is this equilibrium stable? (c) Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b). What minimum energy must be added to the molecule to dissociate it - that is, to separate the two atoms to an infinite distance apart? This is called the dissociation energy of the molecule. (d) For the molecule CO, the cquilibrium distance between the carbon and oxygen atoms is \(1.13 \times 10^{-10} \mathrm{~m}\) and the dissociation energy is \(1.54 \times 10^{-18} \mathrm{~J}\) per molecule. Find the values of the constants \(a\) and \(b\).

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