Question

A mass $m$ is attached to a spring which is held stretched a distance $x$ by a force F, Fig. 6-45, and then released. The spring pulls the mass to the left, towards its natural equilibrium length. Assuming there is no friction, determine the speed of the mass m when the spring returns: (a) to its normal length ($x =$ 0); (b) to half its original extension($x$/2). FIGURE 6-45(Figure Cant copy)

Name: Problem 75, Chapter 6: Physics: Principles with Applications
Uploaded: 2020-08-22T03:48:36-08:00
Duration: 8 min 12 s
Channel: Mark Scythian
Description: A mass m is attached to a spring which is held stretched a distance x by a force F, Fig. 6-45, and then released. The spring pulls the mass to the left, towards its natural equilibrium length. Assuming there is no friction, determine the speed of the mass m when the spring returns: (a) to its normal length (x = 0); (b) to half its original extension(x/2). FIGURE 6-45(Figure Cant copy)

   A mass $m$ is attached to a spring which is held stretched a distance $x$ by a force F, Fig. 6-45, and then released. The spring pulls the mass to the left, towards its natural equilibrium length. Assuming there is no friction, determine the speed of the mass m when the spring returns: (a) to its normal length ($x =$ 0); (b) to half its original extension($x$/2). FIGURE 6-45(Figure Cant copy)

Physics: Principles with Applications

Douglas C. Giancoli 7th Edition

Chapter 6, Problem 75

Instant Answer

Solved by Expert Mark Scythian

08/22/2020

Step 1

This can be written as $F = kx$, where $k$ is the spring constant and $x$ is the displacement. We can solve for $k$ to get $k = \frac{F}{x}$. Show more…

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A mass $m$ is attached to a spring which is held stretched a distance $x$ by a force F, Fig. 6-45, and then released. The spring pulls the mass to the left, towards its natural equilibrium length. Assuming there is no friction, determine the speed of the mass m when the spring returns: (a) to its normal length ($x =$ 0); (b) to half its original extension($x$/2). FIGURE 6-45(Figure Cant copy)

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Key Concepts

Conservation of Mechanical Energy

In frictionless systems, the total mechanical energy (the sum of kinetic energy and potential energy) remains constant. This concept allows us to equate the energy stored in the spring when it is extended with the kinetic energy of the mass when the spring returns toward or passes the equilibrium position, thereby facilitating the calculation of speed at different positions.

Hooke's Law and Spring Potential Energy

Hooke's Law states that the force exerted by a spring is proportional to its displacement from its natural length, leading to the formulation of the spring’s potential energy as (1/2)kx². Understanding this relationship is crucial, as it provides the basis for determining how the potential energy stored in the spring transforms into kinetic energy during the motion of the mass.

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