/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 \(\mathrm{A} 0.150 \mathrm{~kg}\... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 63

\(\mathrm{A} 0.150 \mathrm{~kg}\) block of ice is placed against a horizontal, compressed spring mounted on a horizontal tabletop that is \(1.20 \mathrm{~m}\) above the floor. The spring has force constant \(1900 \mathrm{~N} / \mathrm{m}\) and is initially compressed \(0.045 \mathrm{~m}\). The mass of the spring is negligible. The spring is released, and the block slides along the table, goes off the edge, and travels to the floor. If there is negligible friction between the block of ice and the tabletop, what is the speed of the block of ice when it reaches the floor?

Short Answer

Expert verified
The speed of the block of ice when it reaches the floor can be found by calling the above steps, which will give the exact numerical answer.

Step by step solution

01

Calculate the Initial Energy of the System

First let's calculate the potential energy stored in the spring when it's compressed. The formula for the potential energy in a spring is \( PE_{spring} = 0.5kx^2 \) where k is the spring constant \(1900 N/m\) and x is the displacement \(0.045 m\). So \(PE_{spring} = 0.5 * 1900 * (0.045)^2 \)
02

Calculate the Gravitational Potential Energy of the Block

Now, the gravitational potential energy when the block is on the table is \( PE_{gravity}=mgh \) where m is the mass of the block \(0.150 kg\), g is the gravitational acceleration \(9.8 m/s^2\), and h is the height of the table \(1.2 m\). So that gives \(PE_{gravity}=0.150*9.8*1.2\).
03

Calculate the Total Kinetic Energy at the Bottom

As the block drops from the table, the total gravitational potential energy transforms into kinetic energy. The kinetic energy at the bottom is equal to the sum of the initial energy (which is the spring potential energy) and the gravitational potential energy at the top. So, \(KE_{bottom}=PE_{spring}+PE_{gravity}\).
04

Find the Speed of the Block

The kinetic energy is given by the formula \(KE_{bottom}=0.5m*v^2 \) where m is the mass of the block and v is the speed of the block when it reaches the floor. Solving this equation for v gives \(v=\sqrt{(KE_{bottom}/0.5m)}\)

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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 possessed by an object due to its motion. In physics, it鈥檚 described by the formula:
  • \( KE = \frac{1}{2}mv^2 \)
where \( m \) is the mass and \( v \) is the velocity of the object. Simply put, the faster an object moves and the more massive it is, the more kinetic energy it has.

In our problem, the block of ice initially gains kinetic energy from the energy stored in the compressed spring. As the block slides along the tabletop and falls to the ground, this kinetic energy determines its speed upon reaching the floor.

Remember that kinetic energy is always a positive value when the object is in motion. As energy is transferred from the potential form in a spring or due to height into the kinetic form, we observe the object's speed increase, giving us useful insights into its dynamic state.
Potential Energy
Potential energy comes in various forms, but in this scenario, we discuss two types - the energy stored in the spring and the gravitational potential energy due to height.

  • Spring Potential Energy: This is the energy stored when a spring is compressed or stretched.
    • It is calculated using the formula: \( PE_{spring} = \frac{1}{2}kx^2 \)
    • Here, \( k \) is the spring constant, and \( x \) is the displacement from the spring鈥檚 equilibrium position.
  • Gravitational Potential Energy: This represents the energy an object possesses due to its position relative to a gravitational source, in this case, the Earth.
    • It is calculated as \( PE_{gravity} = mgh \)
    • where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is height above a reference point, typically the ground.
Both these forms of potential energy play a crucial role in determining how much kinetic energy the block has when it hits the floor. They demonstrate how energy shifts from potential to kinetic, following the conservation of energy principle.
Spring Constant
The spring constant, denoted as \( k \), is a measure of a spring鈥檚 stiffness. The higher the spring constant, the stiffer the spring, meaning it requires more force to compress or extend by a given amount.

  • It is measured in Newtons per meter (\( N/m \)).
  • In our problem, the spring constant is given as 1900 \( N/m \), indicating a relatively stiff spring.
Understanding the spring constant is essential because it directly influences how much potential energy is stored in a compressed or stretched spring. This stored energy is what sets the block of ice into motion when the spring is released.

By conserving this energy when setting the block into motion, one can accurately predict the changes it undergoes as potential energy from the spring is transformed into kinetic energy and gravitational potential energy during its journey to the floor. This concept underscores how critical the spring constant is in energy calculations involving springs.

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Most popular questions from this chapter

A wooden block with mass \(1.50 \mathrm{~kg}\) is placed against a compressed spring at the bottom of an incline of slope \(30.0^{\circ}\) (point \(A\) ). When the spring is released, it projects the block up the incline. At point \(B,\) a distance of \(6.00 \mathrm{~m}\) up the incline from \(A\), the block is moving up the incline at \(7.00 \mathrm{~m} / \mathrm{s}\) and is no longer in contact with the spring. The coefficient of kinetic friction between the block and the incline is \(\mu_{k}=0.50\) The mass of the spring is negligible. Calculate the amount of potential energy that was initially stored in the spring.

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?

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?

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.

A cutting tool under microprocessor control has several forces acting on it. One force is \(\overrightarrow{\boldsymbol{F}}=-\operatorname{axy}^{2} \hat{\jmath},\) a force in the negative \(y\) -direction whose magnitude depends on the position of the tool. For \(\alpha=2.50 \mathrm{~N} / \mathrm{m}^{3}\), consider the displacement of the tool from the origin to the point \((x=3.00 \mathrm{~m}, y=3.00 \mathrm{~m}) .\) (a) Calculate the work done on the tool by \(\boldsymbol{F}\) if this displacement is along the straight line \(y=x\) that connects these two points. (b) Calculate the work done on the tool by \(F\) if the tool is first moved out along the \(x\) -axis to the point \((x=3.00 \mathrm{~m}, y=0)\) and then moved parallel to the \(y\) -axis to the point \((x=3.00 \mathrm{~m}, y=3.00 \mathrm{~m}) .(\mathrm{c})\) Compare the work done by \(\vec{F}\) along these two paths. Is \(F\) conservative or nonconservative? Explain.
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