Glossary
Conservative Force
A force for which the work done on an object is independent of the path taken, depending only on the initial and final positions.
Example:
Gravity is a conservative force; the work done by gravity on a ball thrown upwards and caught at the same height is zero, regardless of its trajectory.
Dissipative Forces
Forces that cause mechanical energy to be lost from a system, typically converting it into other forms like heat.
Example:
When a car brakes, the dissipative forces of friction between the brake pads and rotors convert kinetic energy into heat.
Elastic Potential Energy
The energy stored in a deformable elastic object, such as a spring, when it is stretched or compressed from its equilibrium position.
Example:
A toy dart gun stores elastic potential energy in its spring when cocked, which is then converted to kinetic energy to propel the dart.
Equilibrium
A state where the net force acting on an object is zero, often identified on a potential energy graph where the slope is zero.
Example:
A ball resting at the bottom of a bowl is in a stable equilibrium position, as any small displacement results in a restoring force.
Force as Negative Gradient of Potential Energy
A mathematical relationship stating that a conservative force can be found by taking the negative derivative (or gradient) of the potential energy function with respect to position (F = -dU/dr).
Example:
By analyzing a potential energy graph for a particle, you can determine the force as the negative gradient of potential energy at any point, indicating the direction and magnitude of the force.
Gravitational Force
The attractive force between any two objects with mass, which is a fundamental conservative force.
Example:
The gravitational force keeps satellites in orbit around Earth and causes objects to fall downwards.
Gravitational Potential Energy (General Case / Large Distances)
The potential energy of a system of two masses due to their mutual gravitational attraction, defined as U = -Gm1m2/r, with zero potential energy at infinite separation.
Example:
Calculating the energy required to launch a rocket into deep space involves considering the gravitational potential energy (general case / large distances) between the rocket and Earth.
Gravitational Potential Energy (Near Earth)
The potential energy an object possesses due to its position in a uniform gravitational field near the Earth's surface, given by ΔU = mgΔh.
Example:
Lifting a book from the floor to a shelf increases its gravitational potential energy (near Earth).
Hooke's Law
A principle stating that the force needed to extend or compress a spring by some distance is proportional to that distance, given by F_s = -kΔx.
Example:
When you pull back the string of a bow, the force exerted by the bowstring follows Hooke's Law, storing energy for the arrow's launch.
Newton's Law of Universal Gravitation
A law stating that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Example:
The orbit of the Moon around the Earth is governed by Newton's Law of Universal Gravitation.
Path Independence
A characteristic of conservative forces where the work done by the force on an object moving between two points does not depend on the specific path taken.
Example:
When climbing a mountain, the work done by gravity on you depends only on your change in vertical height, demonstrating path independence.
Potential Energy Wells
Local minimums on a potential energy versus position graph, representing stable equilibrium points where a particle can oscillate.
Example:
An atom in a molecule can be thought of as sitting in a potential energy well, oscillating around its stable bond length.
Restoring Force
A force that always acts to bring a system back to its equilibrium position, often seen in springs.
Example:
The force that pulls a pendulum bob back towards its lowest point after being displaced is a restoring force.
Spring Constant (k)
A measure of the stiffness of a spring, indicating how much force is required to stretch or compress it by a unit distance.
Example:
A car's suspension system uses springs with a specific spring constant to absorb shocks and provide a smooth ride.
Spring Force
The force exerted by a spring when it is compressed or stretched, acting to restore the spring to its equilibrium length.
Example:
The spring force in a pogo stick pushes the rider upwards after the spring is compressed.
Total Mechanical Energy
The sum of the kinetic energy and all forms of potential energy within a system, which is conserved in the absence of non-conservative forces.
Example:
In a frictionless roller coaster, the total mechanical energy of the car remains constant, converting between kinetic and gravitational potential energy.
Work-Potential Energy Relationship
The principle that the work done by a conservative force is equal to the negative change in the system's potential energy (W = -ΔU).
Example:
As a roller coaster car descends, the work done by gravity is positive, and its gravitational potential energy decreases, illustrating the work-potential energy relationship.
Zero Work in Closed Path
A property of conservative forces where the total work done by the force on an object moving along any closed loop is zero.
Example:
If you push a box around a complete circle on a frictionless surface, the net work done by the spring force (if attached) would be zero work in a closed path.