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In physics, a force is responsible for changing the state of motion of an object. From computers to cars, machines perform several functions, and some of these require them to move parts back and forth consistently. One part that is used in many different machines is a simple part that today we know as a spring. If you're looking to learn more about springs, look no further. Let's spring into action, and learn some physics!
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Sign up for freeA spring has negligible mass and exerts a force, when stretched or compressed, that is proportional to:
For a spring-mass system on a frictionless horizontal table, the only force acting on the mass in the direction of displacement is the:
The direction of the restoring force will always be ... to the displacement of the object.
The restoring force acting on the spring-mass system depends on the ... and the:
The spring force is described by:
From the equation that describes the spring force, we can determine the angular frequency of the spring-mass system. Its equation is:
A spring-mass system is said to be in equilibrium if there is no net force acting on the object.
A restoring force is a force acting against the displacement to try and bring the system:
The restoring force is what causes the change in acceleration of an object in simple harmonic motion. It is caused by a displacement from the equilibrium position as the system stores:
A spring collection arranged in series, the equivalent spring constant will be ... than the smallest individual spring constant in the set.
In series, the equivalent spring constant will be equal to the sum of the individual spring constants.
A spring has negligible mass and exerts a force, when stretched or compressed, that is proportional to:
For a spring-mass system on a frictionless horizontal table, the only force acting on the mass in the direction of displacement is the:
The direction of the restoring force will always be ... to the displacement of the object.
The restoring force acting on the spring-mass system depends on the ... and the:
The spring force is described by:
From the equation that describes the spring force, we can determine the angular frequency of the spring-mass system. Its equation is:
A spring-mass system is said to be in equilibrium if there is no net force acting on the object.
A restoring force is a force acting against the displacement to try and bring the system:
The restoring force is what causes the change in acceleration of an object in simple harmonic motion. It is caused by a displacement from the equilibrium position as the system stores:
A spring collection arranged in series, the equivalent spring constant will be ... than the smallest individual spring constant in the set.
In series, the equivalent spring constant will be equal to the sum of the individual spring constants.
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A spring has negligible mass and exerts a force, when stretched or compressed, that is proportional to the displacement from its relaxed length. When you grab an object attached to a spring, pull it a distance from its equilibrium position, and release it, the restoring force will pull the object back to equilibrium. For a spring-mass system on a horizontal table, the only force acting on the mass in the direction of displacement is the restoring force exerted by the spring. Using Newton's Second Law, we can set up an equation for the motion of the object. The direction of the restoring force will always be opposite and antiparallel to the displacement of the object. The restoring force acting on the spring-mass system depends on the spring constant and the object's displacement from the equilibrium position.
Fig. 1 - Representation of a spring-mass system, where the mass oscillates about an equilibrium position.
$$\vec{F_{\text{net}}}=m\vec a$$
Along the direction of displacement \(\widehat x\):
$$-kx=m\frac{\operatorname d^2x}{\operatorname dt^2}$$
$$\frac{\operatorname d^2x}{\operatorname dt^2}=-\frac km x$$
Where \(m\) is the mass of the object at the end of the spring in kilograms \((\mathrm{kg})\), \(a_x\) is the acceleration of the object on the \(\text{x-axis}\) in meters per second squared \((\frac{\mathrm m}{\mathrm s^2})\), \(k\) is the spring constant that measures the stiffness of the spring in newtons per meter \((\frac{\mathrm N}{\mathrm m})\), and \(x\) is the displacement in meters \((\mathrm m)\).
This relationship is also known as Hooke's Law, and can be proven by setting up a spring system with hanging masses. Every time that you add a mass, you measure the extension of the spring. If the procedure is repeated, it will be observed that the extension of the spring is proportional to the restoring force, in this case, the weight of the hanging masses.
The above expression looks a lot like the differential equation for simple harmonic motion, so the spring-mass system is a harmonic oscillator, where its angular frequency can be expressed in the below equation.
$$\omega^2=\frac km$$
$$\omega=\sqrt{\frac km}$$
A \(12\;\mathrm{cm}\) spring has a spring constant of \(400\;{\textstyle\frac{\mathrm N}{\mathrm m}}\). How much force is required to stretch the spring to a length of \(14\;\mathrm{cm}\)?
The displacement has a magnitude of
$$x=14\;\mathrm{cm}\;-\;12\;\mathrm{cm}=2\;\mathrm{cm}=0.02\;\mathrm m$$
The spring force has a magnitude of
$$F_s=kx=(400\;{\textstyle\frac{\mathrm N}{\mathrm m}})(0.02\;\mathrm m)=8\;\mathrm N$$
A spring-mass system is said to be in equilibrium if there is no net force acting on the object. This can happen when the magnitude and direction of the forces acting on the object are perfectly balanced, or simply because no forces are acting on the object. Not all forces try to restore the object back to equilibrium, but forces that do so are called restoring forces, and the spring force is one of them.
A restoring force is a force acting against the displacement to try and bring the system back to equilibrium. This type of force is responsible for generating oscillations and is necessary for an object to be in simple harmonic motion. Furthermore, the restoring force is what causes the change in acceleration of an object in simple harmonic motion. As the displacement increases, the stored elastic energy increases and the restoring force increases.
In the diagram below, we see a complete cycle that begins when the mass is released from point \(\text{A}\). The spring forces cause the mass to pass through the equilibrium position all the way up to \(\text{-A}\), just to pass again through the equilibrium position and reach point \(\text{A}\) to complete an entire cycle.
Fig. 2 - Complete oscillation cycle of a spring-mass system.
A collection of springs may act as a single spring, with an equivalent spring constant which we will call \(k_{\text{eq}}\). The springs may be arranged in series or in parallel. The expressions for \(k_{\text{eq}}\) will vary depending on the type of type of arrangement. In series, the inverse of the equivalent spring constant will be equal to the sum of the inverse of the individual spring constants. It is important to note that in an arrangement in series, the equivalent spring constant will be smaller than the smallest individual spring constant in the set.
$$\frac1{k_{eq\;series}}=\sum_n\frac1{k_n}$$
Fig. 3 - Two springs in series.
A set of 2 springs in series have springs constants of \(1{\textstyle\frac{\mathrm N}{\mathrm m}}\) and \(2{\textstyle\frac{\mathrm N}{\mathrm m}}\). What is the value for the equivalent spring constant?
$$\frac1{k_{eq\;series}}=\frac1{1\frac{\mathrm N}{\mathrm m}}+\frac1{2\frac{\mathrm N}{\mathrm m}}$$
$$\frac1{k_{eq\;series}}=\frac32{\textstyle\frac{\mathrm m}{\mathrm N}}$$
$$k_{eq\;series}=\frac23{\textstyle\frac{\mathrm N}{\mathrm m}}$$
In parallel, the equivalent spring constant will be equal to the sum of the individual spring constants.
$$k_{eq\;parallel}=\sum_nk_n$$
Fig. 4 - Two springs in parallel.
A set of 2 springs in parallel have springs constants of \(1{\textstyle\frac{\mathrm N}{\mathrm m}}\) and \(2{\textstyle\frac{\mathrm N}{\mathrm m}}\). What is the value for the equivalent spring constant?
$$k_{eq\;parallel}=1\;{\textstyle\frac{\mathrm N}{\mathrm m}}+\;2{\textstyle\frac{\mathrm N}{\mathrm m}}=3\;{\textstyle\frac{\mathrm N}{\mathrm m}}$$
We can plot the spring force as a function of position and determine the area under the curve. Performing this calculation will provide us with the work done on the system by the spring force and the difference in potential energy stored in the spring due to its displacement. Because in this case, the work done by the spring force depends only on initial and final positions, and not on the path between them, we can derive the change in potential energy from this force. These types of forces are called conservative forces.
Using calculus, we can determine the change in potential energy.
$$\begin{array}{rcl}\triangle U&=&-\int_i^f{\overset\rightharpoonup F}_{cons}\cdot\overset\rightharpoonup{dx},\\\triangle U&=&-\int_i^f\left|{\overset\rightharpoonup F}_{\mathrm{cons}}\right|\left|\overset\rightharpoonup{dx}\right|\cos\left(180^\circ\right),\\\triangle U&=&-\int_i^f\left(kx\right)\left(\mathrm dx\right)\cos\left(180^\circ\right),\\\triangle U&=&\frac12kx_{\mathrm f}^2-\frac12kx_{\mathrm i}^2.\end{array}$$
Fig. 5 - Force vs Displacement graph, the spring constant is the slope and the potential energy is the area below the curve.
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What is an example of a spring force?
An example is spring-mass system in a horizontal table. When you grab an object attached to a spring, pull it a distance from its equilibrium position, and release it, the spring force will pull the object back to equilibrium.
What is spring force formula?
The spring force formular is described by Hooke's Law, F=-kx.
What type of force is spring force?
The spring force is a contact force and a restoring force that is also conservative. There is an interaction between the spring and the object attached to it. The spring forces restores the object to equilibrium when it is displaced. The work done by the spring only depends on the object's initial and final position.
What is spring force?
The spring force is a restoring forced exerted by a spring when it is stretched or compressed. It is proportional and opposite in direction to the displacement from its relaxed length.
Is spring force conservative?
Because in this case, the work done by the spring force depends only on initial and final positions, not on the path between them, the force is called a conservative force.
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