Glossary
Change in Length (Δx)
The displacement of a spring from its relaxed (equilibrium) length, measured as either a stretch (positive) or a compression (negative).
Example:
If a spring is stretched from 10 cm to 15 cm, its change in length (Δx) is 5 cm.
Direction of Spring Force
The spring force always points towards the equilibrium position, acting to restore the spring to its relaxed state.
Example:
When a trampoline is compressed by a jumper, the direction of spring force is upwards, pushing the jumper back into the air.
Displacement vector from equilibrium ($\Delta \vec{x}$)
The vector quantity representing the change in a spring's length or position relative to its natural, unstretched or uncompressed state.
Example:
If a spring is stretched 5 cm from its resting length, its Displacement vector from equilibrium is 5 cm in the direction of the stretch.
Energy Conservation
A fundamental principle stating that the total mechanical energy (kinetic plus potential) of a system remains constant if only conservative forces do work.
Example:
When a mass oscillates on a spring, its kinetic and potential energies continuously convert, but their sum remains constant due to Energy Conservation.
Equilibrium Position
The position where the net force acting on an object is zero, often corresponding to the spring's relaxed length in a horizontal system or the point where spring force balances gravity in a vertical system.
Example:
A mass hanging motionless from a spring has reached its equilibrium position.
Equilibrium position
The natural resting position of a spring where no net force acts upon it, and its potential energy is minimized.
Example:
A spring hanging freely from a ceiling will settle at its Equilibrium position where the gravitational force is balanced by the spring's own force.
Equivalent Spring Constant ($k_{eq}$)
A single spring constant that represents the combined stiffness of multiple springs in a system, simplifying analysis.
Example:
To analyze a complex system with two springs, we can calculate the Equivalent Spring Constant to treat them as one effective spring.
Free-body diagram
A visual representation used to analyze forces acting on an object, showing all forces as vectors originating from the object's center of mass.
Example:
Before solving a problem involving a block on an inclined plane, drawing a Free-body diagram helps identify gravitational, normal, and frictional forces.
Hooke's Law
A fundamental principle stating that the force exerted by an ideal spring is directly proportional to its displacement from equilibrium, always acting in the opposite direction of the displacement.
Example:
When you pull a spring scale to weigh an object, the force you feel is described by Hooke's Law, indicating how much the spring stretches.
Hooke's Law
A fundamental principle stating that the force exerted by an ideal spring is directly proportional to its displacement from equilibrium and acts in the opposite direction, expressed as F = -kΔx.
Example:
Using Hooke's Law, you can calculate the force required to stretch a spring by a specific amount if you know its spring constant.
Ideal Spring
A theoretical spring model assumed to have negligible mass and exert a linear force directly proportional to its displacement from equilibrium.
Example:
In most AP Physics 1 problems, we analyze an ideal spring to simplify calculations and focus on fundamental principles.
Ideal Springs
Theoretical springs with negligible mass that exert a force perfectly proportional to their displacement from equilibrium.
Example:
In a physics problem, we often assume a toy car's suspension uses Ideal Springs to simplify calculations of its oscillation.
Linear Force (of spring)
A characteristic of an ideal spring where the force it exerts is directly proportional to the amount it is stretched or compressed.
Example:
If you stretch a spring twice as far, it will exert twice the linear force according to Hooke's Law.
Negative Sign (in Hooke's Law)
The negative sign in the Hooke's Law formula (F = -kΔx) signifies that the spring force always acts in the direction opposite to the displacement from equilibrium.
Example:
If you stretch a spring to the right (positive Δx), the negative sign indicates the spring force pulls to the left.
Negligible Mass (of spring)
An assumption in ideal spring models where the mass of the spring itself is considered so small that it does not significantly affect the system's dynamics or calculations.
Example:
When calculating the period of a mass-spring system, we assume the spring has negligible mass to simplify the formula.
Non-Ideal Springs
Real-world springs that possess mass and whose force may not be perfectly proportional to displacement due to material properties or operating conditions.
Example:
A car's actual suspension system uses Non-Ideal Springs because their mass and internal friction affect how they respond to bumps.
Relaxed Length
The natural length of a spring when no external forces are acting upon it, and it is neither stretched nor compressed.
Example:
Before hanging any weights, the length of a spring hanging freely is its relaxed length.
Restoring force
A force that always acts to bring a system back to its equilibrium position, opposing any displacement from that position.
Example:
When a pendulum swings, gravity provides a Restoring force that pulls it back towards its lowest point.
Restoring forces
Forces that always act to bring an object back to its equilibrium position, opposing the displacement.
Example:
When you pull a pendulum to the side, gravity acts as a restoring force to bring it back to its lowest point.
Simple Harmonic Motion
A type of periodic motion where the restoring force is directly proportional to the displacement and acts towards the equilibrium position, leading to sinusoidal oscillations.
Example:
A mass attached to an ideal spring, oscillating back and forth without friction, exhibits Simple Harmonic Motion.
Simple Harmonic Motion (SHM)
A type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and always acts towards the equilibrium position, resulting in sinusoidal oscillations.
Example:
A mass oscillating back and forth on an ideal spring is a classic example of simple harmonic motion.
Spring Potential Energy
Energy stored in an elastic material, such as a spring, due to its compression or extension from its equilibrium position, calculated as 1/2 k(Δx)^2.
Example:
A toy dart gun stores spring potential energy when its spring is compressed, which is then converted into the dart's kinetic energy upon release.
Spring constant ($k$)
A measure of the stiffness of a spring, indicating how much force is required to stretch or compress it by a certain unit distance.
Example:
A stiff car suspension spring would have a high Spring constant, meaning it resists deformation strongly.
Spring constant (k)
A measure of the stiffness of a spring, indicating how much force is required to stretch or compress it by a given unit of length (measured in N/m).
Example:
A very stiff spring, like those in a car's suspension, will have a high spring constant.
Spring force vector ($\vec{F}_{s}$)
The vector quantity representing the force exerted by a spring, always directed towards its equilibrium position.
Example:
If you compress a spring, the Spring force vector points outwards, pushing against the compression.
Spring forces
Forces that arise when a spring is stretched or compressed from its natural length, crucial for understanding oscillations and energy storage.
Example:
A car's suspension system relies on spring forces to absorb shocks and provide a smooth ride.
Springs in Parallel
A configuration where springs are connected side-by-side, sharing the same displacement, resulting in a system that is stiffer than any individual spring.
Example:
When two springs are attached to the same block and pull it together, they are configured as Springs in Parallel.
Springs in Series
A configuration where springs are connected end-to-end, resulting in a system that is more compliant (easier to stretch) than any individual spring.
Example:
If you hang a weight from one spring, and then attach another spring below the first one to the weight, the springs are connected in Springs in Series.