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Chapter 7: Problem 25

You are asked to design a spring that will give a \(1160 \mathrm{~kg}\) satellite a speed of \(2.50 \mathrm{~m} / \mathrm{s}\) relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of \(5.00 \mathrm{~g}\). The spring's mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will all be negligible. (a) What must the force constant of the spring be? (b) What distance must the spring be compressed?

Short Answer

Expert verified
The force constant of the spring should be \( \frac{56840 N}{x} \). The spring must be compressed by an amount determined by solving the equation \( 3625 J = \frac{1}{2} * \frac{56840 N}{x} * x^2 \)

Step by step solution

01

Find the Kinetic Energy

The kinetic energy (KE) can be calculated by the formula \( KE = \frac{1}{2} m v^2 \) where m is the mass and v is the speed. So, \( KE = \frac{1}{2} * 1160 kg * (2.5 m/s)^2 = 3625 J \)
02

Determine the Force Constant

Start by first calculating the net force (F) on the satellite using the equation \( F = m a \) where m is the mass and a is the acceleration. Given that acceleration is \( 5.00 g = 5 \times 9.8 m/s^2 = 49 m/s^2 \), so the net force is \( F = 1160 kg * 49 m/s^2 = 56840 N \). As the spring obeys Hooke's law, Force is equal to kx, where k is the spring constant and x is the spring displacement. When the spring is at its maximum compression, the satellite is momentarily at rest and thus acceleration is maximum. Hence \( x = \frac{F_{max}}{k} \). Substituting acceleration for force, we get \( k = \frac{F_{max}}{x} \). So, \( k = \frac{56840 N}{x} \)
03

Determine the Spring Compression

The work done to compress the spring (which is equal to the kinetic energy of the satellite) is given by the elastic potential energy stored in the spring \( U = \frac{1}{2} k x^2 \) where k is the spring constant and x is the spring displacement. Plugging in the kinetic energy found in step 1 and the spring constant determined in step 2, and solving for x you get: \( 3625 J = \frac{1}{2} * \frac{56840 N}{x} * x^2 \). Solving for the displacement x, we get the distance the spring should be compressed.

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Key Concepts

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

Kinetic Energy
Kinetic energy is the energy that an object possesses due to its motion. It is one of the most fundamental concepts in physics and is described by the formula:\[ KE = \frac{1}{2} m v^2 \]Where:
  • \( m \) is the mass of the object.
  • \( v \) is the velocity (or speed) of the object.
In this exercise, we calculated the kinetic energy of a satellite with a mass of 1160 kg moving at 2.50 m/s. By substituting these values into the equation, we found that the kinetic energy is 3625 Joules.
This energy is crucial because it tells us how much energy is available to be converted from potential energy stored in the spring when it is compressed and released.
Hooke's Law
Hooke's Law describes the behavior of springs and states that the force needed to extend or compress a spring is proportional to the distance it is stretched or compressed. The formula is expressed as:\[ F = kx \]Where:
  • \( F \) is the force exerted by the spring.
  • \( k \) is the spring constant, which measures the stiffness of the spring.
  • \( x \) is the displacement from the equilibrium position.
In our problem, this law is used to relate the force the spring exerts on the satellite with how much the spring is compressed. The satellite must be designed in such a way that, even at maximum compression, Hooke鈥檚 Law is fulfilled entirely.The force and displacement relationship enables us to solve for other variables, such as the spring constant and spring displacement, which are important for understanding how far a spring needs to be compressed to achieve the desired kinetic energy.
Spring Constant
The spring constant, symbolized as \( k \), is a measure of a spring's stiffness. The larger the spring constant, the stiffer the spring, which means more force is required to compress or extend it by a given amount. In this exercise, the spring constant is crucial for designing a spring that provides the necessary force to launch a satellite at a specified speed. Using Hooke鈥檚 Law, we determined that the spring constant can be calculated with the formula:\[ k = \frac{F_{max}}{x} \]Here:
  • \( F_{max} \) is the maximum force exerted by the spring when compressed.
  • \( x \) is the compression distance of the spring.
Determining the spring constant is an essential step in ensuring that the spring can deliver the required acceleration to the satellite without exceeding its structural limits.
Elastic Potential Energy
Elastic potential energy is the energy stored in elastic materials, such as springs, when they are compressed or stretched. It can be calculated using the formula:\[ U = \frac{1}{2} k x^2 \]Where:
  • \( U \) is the elastic potential energy.
  • \( k \) is the spring constant.
  • \( x \) is the displacement from the equilibrium position.
This concept comes into play when we want to determine how much energy is stored in a compressed or stretched spring, which in turn can be converted into kinetic energy. In the problem, the spring's compression allows for the satellite's kinetic energy of 3625 Joules to be achieved. By knowing the kinetic energy, we could trace back to find the required compression distance as well as ensure enough energy is being stored to reach the desired velocity of the satellite.

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Most popular questions from this chapter

DATA You are designing a pendulum for a science museum. The pendulum is made by attaching a brass sphere with mass \(m\) to the lower end of a long. light metal wire of (unknown) length \(L\). A device near the top of the wire measures the tension in the wire and transmits that information to your laptop computer. When the wire is vertical and the sphere is at rest, the sphere's center is \(0.800 \mathrm{~m}\) above the floor and the tension in the wire is \(265 \mathrm{~N}\). Keeping the wire taut, you then pull the sphere to one side (using a ladder if necessary) and gently release it. You record the height \(h\) of the center of the sphere above the floor at the point where the sphere is released and the tension \(T\) in the wire as the sphere swings through its lowest point. You collect your results: \begin{tabular}{l|lllllll} \(h(\mathbf{m})\) & 0.800 & 2.00 & 4.00 & 6.00 & 8.00 & 10.0 & 12.0 \\ \hline \(\boldsymbol{T}(\mathrm{N})\) & 265 & 274 & 298 & 313 & 330 & 348 & 371 \end{tabular} Assume that the sphere can be treated as a point mass, ignore the mass of the wire, and assume that total mechanical energy is conserved through each measurement. (a) Plot \(T\) versus \(h,\) and use this graph to calculate \(L\). (b) If the breaking strength of the wire is \(822 \mathrm{~N}\), from what maximum height \(h\) can the sphere be released if the tension in the wire is not to exceed half the breaking strength? (c) The pendulum is swinging when you leave at the end of the day. You lock the museum doors, and no one enters the building until you return the next morning. You find that the sphere is hanging at rest. Using energy considerations, how can you explain this behavior?

A small rock with mass \(m\) is released from rest at the inside rim of a large, hemispherical bowl (point \(A\) ) that has radius \(R\), as shown in Fig. E7.9. If the normal force exerted on the rock as it slides through its lowest point (point \(B\) ) is twice the weight of the rock, how much work did friction do on the rock as it moved from \(A\) to \(B ?\) Express your answer in terms of \(m, R,\) and \(g\)

You are designing a delivery ramp for crates containing exercise equipment. The \(1470 \mathrm{~N}\) crates will move at \(1.8 \mathrm{~m} / \mathrm{s}\) at the top of a ramp that slopes downward at \(22.0^{\circ} .\) The ramp exerts a \(515 \mathrm{~N}\) kinctic friction force on cach crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of \(5.0 \mathrm{~m}\) along the ramp. Once stopped, a crate must not rebound back up the ramp. Calculate the largest force constant of the spring that will be needed to meet the design criteria.

If a fish is attached to a vertical spring and slowly lowered to its equilibrium position, it is found to stretch the spring by an amount \(d\). If the same fish is attached to the end of the unstretched spring and then allowed to fall from rest, through what maximum distance does it stretch the spring? (Hint: Calculate the force constant of the spring in terms of the distance \(d\) and the mass \(m\) of the fish.)

\(\mathrm{CALC}\) A \(3.00 \mathrm{~kg}\) fish is attached to the lower end of a vertical spring that has negligible mass and force constant \(900 \mathrm{~N} / \mathrm{m}\). The spring initially is neither stretched nor compressed. The fish is released from rest. (a) What is its speed after it has descended \(0.0500 \mathrm{~m}\) from its initial position? (b) What is the maximum speed of the fish as it descends?
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