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Chapter 13: Problem 3

The force constant of a spring is \(137 \mathrm{~N} / \mathrm{m}\). Find the magnitude of the force required to (a) compress the spring by \(4.80 \mathrm{~cm}\) from its unstretched length and (b) stretch the spring by \(7.36 \mathrm{~cm}\) from its unstretched length.

Short Answer

Expert verified
The magnitude of the force required to compress the spring by 4.80 cm from its unstretched length is 6.576 N. The magnitude of the force required to stretch the spring by 7.36 cm from its unstretched length is 10.0832 N.

Step by step solution

01

Understanding Hooke's Law

Hooke's law states that the force \(F\) needed to extend or compress a spring by some distance \(x\) is proportional to that displacement. This can be mathematically expressed as: \(F = kx\), where \(k\) is the spring constant.
02

Applying Hooke's Law (Part A)

Substitute the given values into Hooke's law: \(F = kx\). The spring constant \(k\) is given as \(137 \mathrm{~N} / \mathrm{m}\), and the displacement \(x\) is \(4.80 \mathrm{~cm}\) which we'll need to convert into meters for consistency. Now, \(F = 137 \times 0.048\).
03

Calculation (Part A)

Perform the multiplication to find the force: \(F = 6.576 \mathrm{~N}\). This is the magnitude of the force required to compress the spring by \(4.80 \mathrm{cm}\) from its unstretched length.
04

Applying Hooke's Law (Part B)

We will repeat step 2, this time for a displacement of \(7.36 \mathrm{~cm}\). Convert this displacement into meters and substitute into Hooke's law: \(F = 137 \times 0.0736\).
05

Calculation (Part B)

Perform the multiplication to find the force: \(F = 10.0832 \mathrm{~N}\). This is the magnitude of the force required to stretch the spring by \(7.36 \mathrm{~cm}\) from its unstretched length.

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Key Concepts

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

Spring Constant
When dealing with springs, understanding the spring constant is key. The spring constant, denoted as \( k \), measures a spring's stiffness. The stiffer the spring, the higher the value of \( k \). This means more force is required to compress or stretch the spring by a given distance. In physics, the spring constant is expressed in Newton per meter (N/m), representing the force needed per unit of length. For instance, if the spring constant \( k \) is 137 N/m, it signifies that 137 Newtons of force are needed to compress or extend the spring by 1 meter.
  • The spring constant is unique to each spring.
  • Its value gives an intuition about the force requirements.
  • Higher \( k \) values indicate stiffer springs.
Understanding this concept allows you to grasp how different springs react to various forces and helps apply Hooke's Law efficiently.
Force Calculation
Force calculation using Hooke's Law is quite straightforward. Hooke's Law is used to determine the force applied to a spring when it is stretched or compressed. The formula \( F = kx \) correlates the force \( F \) with the spring constant \( k \) and the displacement \( x \). First, ensure that the displacement is measured in meters to maintain the correct unit consistency. For example, our problem involves a spring constant \( k = 137 \) N/m, and displacements of 4.80 cm and 7.36 cm. These displacements convert into meters as 0.048 m and 0.0736 m, respectively.
  • The force is directly proportional to both displacement and spring constant.
  • Ensure all measurements are in compatible units before calculation.
By plugging these values into the formula, we calculate the necessary force to compress or stretch the spring. This methodology helps assess how much force different spring actions require.
Displacement in Meters
When working with Hooke's Law, emphasis on displacement measurement is crucial. Displacement refers to the change in length from the spring's natural, unstressed state. In most physics problems, it's essential to express displacement in meters. Doing so maintains unit consistency, making it compatible with the spring constant's units of Newton per meter (N/m). In the example provided, initially given in centimeters (cm), we convert these to meters by dividing by 100 since 1 meter = 100 centimeters. This simple step ensures the formula application is correct.
  • Always convert displacement to meters.
  • Accurate conversion is key for precise force calculations.
This method is an essential practice in physics, ensuring accurate calculations and successful solutions.

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Most popular questions from this chapter

A simple pendulum is \(5.00 \mathrm{~m}\) long. (a) What is the period of simple harmonic motion for this pendulum if it is located in an elevator accelerating upward at \(5.00 \mathrm{~m} / \mathrm{s}^{2}\) ? (b) What is its period if the elevator is accelerating downward at \(5.00 \mathrm{~m} / \mathrm{s}^{2}\) ? (c) What is the period of simple harmonic motion for the pendulum if it is placed in a truck that is accelerating horizontally at \(5.00 \mathrm{~m} / \mathrm{s}^{2}\) ?

A harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes \(40.0\) vibrations in \(30.0 \mathrm{~s}\). Also, a given maximum travels \(425 \mathrm{~cm}\) along the rope in \(10.0 \mathrm{~s}\). What is the wavelength?

The position of a \(0.30-\mathrm{kg}\) object attached to a spring is described by $$ x=(0.25 \mathrm{~m}) \cos (0.4 \pi t) $$ Find (a) the amplitude of the motion, (b) the spring constant, (c) the position of the object at \(t=0.30 \mathrm{~s}\), and (d) the object's speed at \(t=0.30 \mathrm{~s}\).

An astronaut on the Moon wishes to measure the local value of \(g\) by timing pulses traveling down a wire that has a large object suspended from it. Assume a wire of mass \(4.00 \mathrm{~g}\) is \(1.60 \mathrm{~m}\) long and has a \(3.00-\mathrm{kg}\) object suspended from it. A pulse requires \(36.1 \mathrm{~ms}\) to traverse the length of the wire. Calculate \(g_{\mathrm{Mon}}\) from these data. (You may neglect the mass of the wire when calculating the tension in it.)

A cart of mass \(250 \mathrm{~g}\) is placed on a frictionless horizontal air track. A spring having a spring constant of \(9.5 \mathrm{~N} / \mathrm{m}\) is attached between the cart and the left end of the track. If the cart is displaced \(4.5 \mathrm{~cm}\) from its equilibrium position, find (a) the period at which it oscillates, (b) its maximum speed, and (c) its speed when it is located \(2.0 \mathrm{~cm}\) from its equilibrium position.
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