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Chapter 12: Problem 95

Two identical springs are connected parallel to one another; that is, they lie side by side. Is the spring constant of the resulting compound spring greater than, less than, or equal to the spring constant of a single spring? Explain.

Short Answer

Expert verified
The spring constant in parallel is greater, equal to twice the single spring's constant.

Step by step solution

01

Understanding Spring Constant in Parallel

When two springs are connected in parallel, they each experience the same displacement when a force is applied. The total spring constant for springs connected in parallel is given by the sum of the individual spring constants.
02

Dynamics of Parallel Springs

Consider two springs with spring constant \( k \) each. When a force \( F \) is applied, both springs compress or extend by the same amount, \( x \). The force exerted by each spring is \( k \times x \).
03

Application of Hooke's Law

According to Hooke's law, the force exerted by a spring is \( F = kx \). For the parallel combination, the forces add up, so the total force \( F_{total} = kx + kx = 2kx \).
04

Determine the Resulting Spring Constant

The resulting force equation for the parallel springs is \( F_{total} = (k_{total})x \). Therefore, \( k_{total}x = 2kx \), which implies \( k_{total} = 2k \).
05

Conclusion

The spring constant of the compound spring in parallel, \( k_{total} \), is equal to twice the spring constant of a single spring, indicating that it is greater.

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Key Concepts

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

Parallel Springs
When two springs are arranged in parallel, they each endure the same amount of displacement in response to an applied force. This setup means that the springs work together to counteract any force, effectively doubling their strength. Imagine this as two sets of muscles working side by side. Each spring exerts its own force according to its spring constant. When connected in parallel, these forces summate, resulting in a stronger overall reaction.
  • The effective spring constant for parallel springs is simply the sum of the individual spring constants.
  • For example, if each spring has a spring constant of \( k \), the total spring constant when they are parallel would be \( 2k \).
Understanding that the spring constant increases in parallel configurations is crucial for practical applications such as designing suspension systems where enhanced stiffness is required.
Hooke's Law
Hooke鈥檚 Law is fundamental to understanding how springs behave under force. It states that the force \( F \) exerted by a spring is directly proportional to its displacement \( x \), and this relationship is mediated by the spring constant \( k \). This can be expressed mathematically as: \[ F = kx \]This linear relationship implies that the stiffer the spring (a larger \( k \)), the more force is required to displace it by a given amount. This makes Hooke's Law pivotal in calculating the total force within systems like parallel springs.
  • In a parallel spring system, the combined spring force adheres to Hooke's Law, summing the contributions from each spring.
  • Therefore, knowing each spring's constant helps predict the system's behavior under different displacements.
Spring Force
The spring force is the force exerted by a spring on any load attached to it, as it attempts to restore its original shape. This force is symmetric and opposite to the displacement direction. In the example of parallel springs, each spring contributes to the total force that resists displacement.
  • The spring force for an individual spring is given by \( F = kx \).
  • For parallel springs, you calculate the total force by summing up the individual forces exerted by each spring.
The highlighted concept is that in a parallel configuration, you multiply the individual spring force, telling you how effective the system will work against expansion or compression.
Compound Spring System
A compound spring system integrates multiple springs to modify the spring constant, improving or adjusting the system's mechanical properties. When springs are combined, how they are configured鈥攕eries or parallel鈥攄etermines the overall behavior.
In the scenario described with parallel springs:
  • The spring constant doubles, meaning the system is considerably stiffer. This is beneficial in situations where greater resistance to force is needed, such as heavy machinery supports or vehicle suspension systems.
  • Parallel arrangements enhance the load-bearing capacity without changing the displacement length for a given force.
Understanding compound spring systems enables engineers and designers to tailor solutions for specific physical and mechanical needs, optimizing for varying stiffness and flexibility as desired.

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Most popular questions from this chapter

Using a Hydrometer A hydrometer is a device for measuring fluid density. It is constructed as shown in Figure 12.31. If the hydrometer pulls a sample of fluid 1 into it, the small float inside the tube is submerged to the level \(1 .\) When fluid 2 is sampled, the float is submerged to level 2 . Is the density of fluid 1 greater than, less than, or equal to the density Figure \(12.31\) of fluid 2? (This is how a car mechanic tests your car's antifreeze level. Since antifreeze [ethylene glycol] is more dense than water, the higher the density of coolant in your radiator, the more antifreeze protection your car has.)

A spring has a spring constant of \(56 \mathrm{~N} / \mathrm{m}\). How much is the spring compressed by a force of \(6.1 \mathrm{~N}\) ?

A block of wood has a steel ball glued to one surface. The block can float in water with the ball "high and dry" on its top surface. (a) When the block is inverted and the ball is immersed in the water, does the volume of wood that is submerged increase, decrease, or stay the same? (b) Choose the best explanation from among the following: A. When the block is inverted, the ball pulls it downward, causing more of the wood to be submerged. B. The same amount of mass is supported in either case; therefore, the amount of wood that is submerged is the same. C. When the block is inverted, the ball experiences a buoyant force, which reduces the buoyant force that must be provided by the wood.

What happens when a solid is stretched beyond its elastic limit?

Pulling on a spring with a force of \(1.2 \mathrm{~N}\) causes it to stretch by \(6.4 \mathrm{~cm}\). What is the spring constant for this spring?
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