Intro to Springs (Hooke's Law): Videos & Practice Problems

Springs exhibit restoring forces, as described by Hooke's Law, which states that the force exerted by a spring (F_S) is proportional to its deformation (x): F=-kx, where k is the spring constant. The deformation can be compression or extension, and k indicates the stiffness of the spring. In vertical spring systems, equilibrium occurs when the spring force equals the weight of the mass (mg), leading to the equation kx = mg .

Intro to Springs

Understanding springs and spring forces is essential in physics, particularly in mechanics. When a force is applied to a spring, it either compresses or stretches, demonstrating Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. The force exerted by the spring, denoted as \( F_s \), is equal in magnitude but opposite in direction to the applied force \( F_a \). This relationship can be expressed mathematically as:

\( F_s = -k x \)

In this equation, \( k \) represents the spring constant, a measure of the spring's stiffness, while \( x \) is the deformation of the spring from its relaxed position. The negative sign indicates that the spring force acts in the opposite direction to the deformation. The deformation \( x \) is defined as the change in length of the spring, which can be calculated as the absolute value of the final length minus the initial length. Thus, \( x = |L_f - L_i| \), where \( L_f \) is the final length and \( L_i \) is the initial length.

The spring constant \( k \) can also be referred to as the stiffness constant or force constant, and its units are expressed in newtons per meter (N/m). A higher value of \( k \) indicates a stiffer spring that requires more force to achieve the same deformation compared to a spring with a lower \( k \). For example, a thin spring in a pen has a low \( k \), making it easy to compress, while a safety spring in an elevator shaft has a high \( k \) to withstand significant forces without deforming excessively.

When analyzing the forces acting on a spring, it is crucial to recognize that the spring force is a restoring force, always acting to return the spring to its original length. If a spring is stretched or compressed, it will exert a force in the opposite direction to restore itself to equilibrium. This behavior can be observed in both horizontal and vertical spring systems.

In vertical spring systems, when a mass is attached to a spring, the weight of the mass (given by \( mg \), where \( m \) is mass and \( g \) is the acceleration due to gravity) will stretch the spring until it reaches equilibrium. At this point, the forces balance, leading to the equation:

\( kx = mg \

This equation is particularly useful for understanding the behavior of springs in equilibrium situations. It is important to note that this relationship holds true only when the mass is allowed to come to rest slowly, ensuring that the system reaches a stable equilibrium without oscillation.

In summary, the key concepts surrounding springs include the relationship between applied force and spring force, the significance of the spring constant, and the behavior of springs in both horizontal and vertical orientations. Understanding these principles allows for the effective analysis and application of spring mechanics in various physical scenarios.

A vertical spring is originally 60 cm long. When you attach a 5 kg object to it, the spring stretches to 70 cm. 
(a) Find the force constant on the spring.
(b) You now attach an additional 10 kg to the spring. Find its new length.
(For the multiple choice selection, answer only part (b). Use g=10 m/s².)

• A. (b) L_new=80 cm
• B. (b) L_new=90 cm
• C. (b) L_new=100 cm
• D. (b) L_new=110 cm

Answer: (B)

Here’s what students ask on this topic:

What is Hooke's Law and how is it applied to springs?

Hooke's Law states that the force exerted by a spring is directly proportional to its deformation (either compression or extension). The law is mathematically expressed as:

$F=-kx$

where $F$ is the force exerted by the spring, $k$ is the spring constant (a measure of the spring's stiffness), and $x$ is the deformation of the spring from its equilibrium position. The negative sign indicates that the force exerted by the spring is in the opposite direction of the deformation. This law is fundamental in understanding how springs behave under various forces and is widely used in physics and engineering applications.

How do you calculate the spring constant (k) of a spring?

The spring constant $k$ can be calculated using Hooke's Law, which is given by:

$F=-kx$

Rearranging the equation to solve for $k$, we get:

$k=\frac{F}{x}$

Here, $F$ is the force applied to the spring, and $x$ is the deformation of the spring. By measuring the force applied and the resulting deformation, you can calculate the spring constant. The units of $k$ are typically Newtons per meter (N/m).

What is the significance of the negative sign in Hooke's Law?

The negative sign in Hooke's Law, expressed as $F=-kx$, indicates that the force exerted by the spring is in the opposite direction of the deformation. This is a reflection of Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. In the context of springs, if you compress the spring (shorten it), the spring exerts a force to expand back to its original length. Conversely, if you stretch the spring (lengthen it), the spring exerts a force to compress back to its original length. The negative sign is crucial for understanding the restoring nature of spring forces.

How do you determine the deformation (x) of a spring?

The deformation $x$ of a spring is the change in length from its relaxed (equilibrium) position. It can be calculated as:

$x=|L{f}_{}-L{i}_{}|$

where $L{f}_{}$ is the final length of the spring and $L{i}_{}$ is the initial length of the spring. The absolute value is taken to ensure that deformation is always a positive quantity, regardless of whether the spring is compressed or stretched. This change in length is what is used in Hooke's Law to determine the force exerted by the spring.

What is the equilibrium condition for a vertical spring with a mass attached?

For a vertical spring with a mass attached, equilibrium occurs when the spring force equals the gravitational force acting on the mass. This can be expressed as:

$kx=mg$

where $k$ is the spring constant, $x$ is the deformation of the spring, $m$ is the mass, and $g$ is the acceleration due to gravity. At equilibrium, the upward force exerted by the spring balances the downward gravitational force, resulting in no net force and no further movement. This condition is crucial for understanding how springs behave in vertical systems and is often used in problems involving vertical spring-mass setups.