23/08/2025
FEATURES OF A CIRCLE
A circle is defined by its center and radius, with all points on the circle being equidistant from the center. Key properties include its circumference (distance around the circle), the diameter (the longest chord, equal to twice the radius), and that it has infinite lines of symmetry. Other properties involve relationships between elements like chords, tangents, and angles within the circle, such as equal chords being equidistant from the center and tangents being perpendicular to the radius at the point of contact.
MAJOR COMPONENTS
Center:
A fixed point from which all points on the circle's boundary are equidistant.
Radius (r):
A line segment from the center to any point on the circumference.
Diameter (d):
A line segment that passes through the center and connects two points on the circumference. It is the longest chord and is twice the radius (d = 2r).
Circumference (C):
The total distance around the boundary of the circle.
Chord:
A line segment connecting any two points on the circumference.
Tangent:
A line that touches the circle at exactly one point.
SYMMETRY & RELATIONSHIPS
➡️Lines of Symmetry:
A circle has infinite lines of symmetry, with every line through its center being a line of symmetry.
➡️Equal Chords:
Chords of equal length are equidistant from the center of the circle.
➡️Radius and Tangent:
A radius drawn to the point of contact of a tangent line is perpendicular to the tangent line.
➡️Perpendicular Bisector of a Chord:
The perpendicular bisector of any chord of a circle passes through the center of the circle.
OTHER PROPERTIES
Congruent Circles:
Circles with the same radius are congruent.
Similar Circles: All circles are similar to one another.
Angle in a Semicircle:
An angle inscribed in a semicircle is always a right angle (90 degrees).
The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circumference.
23/08/2025
SIMPLE HARMONIC MOTION AND ENERGY CONSERVATION
In Simple Harmonic Motion (SHM), energy is continuously converted between kinetic and potential forms, but the total mechanical energy of the system remains constant (conserved) as long as there are no dissipative forces like friction. At the extreme points of an oscillation, all energy is potential, while at the equilibrium position, all energy is kinetic. This principle allows for the relationship between velocity, position, and total energy to be described by the equation E = Ek + Ep = constant, where Ek is kinetic energy and Ep is potential energy.
ENERGY TRANSFORMATION IN SHM
Potential Energy (Ep):
This is stored energy that depends on the object's position or deformation. For a spring-mass system, it's stored in the spring when it's stretched or compressed.
Kinetic Energy (Ek):
This is the energy of motion, which depends on the object's mass and velocity.
TRANSFORMATION:
As the object moves from its maximum displacement (where Ep is maximum and Ek is zero) towards the equilibrium position, potential energy converts into kinetic energy. As it moves away from equilibrium, kinetic energy converts back into potential energy.
Conservation of Total Energy
Constant Sum:
The total energy (E) is always the sum of the kinetic and potential energies (E = Ek + Ep) and never changes throughout the oscillation.
Proportional to Amplitude:
The total energy is directly proportional to the square of the amplitude (the maximum displacement from equilibrium).
Equation:
The conservation of energy can be expressed as:
At maximum displacement (A),
x = A, v = 0: E = 0 + Ep_max = 1/2 kA²
At the equilibrium position (x = 0):
E = Ek_max + 0
At any intermediate position (x):
E = 1/2 mv² + 1/2 kx²
Therefore, the total energy is constant:
1/2 mv² + 1/2 kx² = 1/2 kA².
SIGNIFICANCE OF ENERGY CONSERVATION
Predicting Behavior:
Applying the energy conservation principle helps predict the system's behavior, like its velo
23/08/2025
SIMPLE HARMONIC MOTION AND ENERGY CONSERVATION
In Simple Harmonic Motion (SHM), energy is continuously converted between kinetic and potential forms, but the total mechanical energy of the system remains constant (conserved) as long as there are no dissipative forces like friction. At the extreme points of an oscillation, all energy is potential, while at the equilibrium position, all energy is kinetic. This principle allows for the relationship between velocity, position, and total energy to be described by the equation E = Ek + Ep = constant, where Ek is kinetic energy and Ep is potential energy.
ENERGY TRANSFORMATION IN SHM
Potential Energy (Ep):
This is stored energy that depends on the object's position or deformation. For a spring-mass system, it's stored in the spring when it's stretched or compressed.
Kinetic Energy (Ek):
This is the energy of motion, which depends on the object's mass and velocity.
TRANSFORMATION:
As the object moves from its maximum displacement (where Ep is maximum and Ek is zero) towards the equilibrium position, potential energy converts into kinetic energy. As it moves away from equilibrium, kinetic energy converts back into potential energy.
Conservation of Total Energy
Constant Sum:
The total energy (E) is always the sum of the kinetic and potential energies (E = Ek + Ep) and never changes throughout the oscillation.
Proportional to Amplitude:
The total energy is directly proportional to the square of the amplitude (the maximum displacement from equilibrium).
Equation:
The conservation of energy can be expressed as:
At maximum displacement (A),
x = A, v = 0: E = 0 + Ep_max = 1/2 kA²
At the equilibrium position (x = 0):
E = Ek_max + 0
At any intermediate position (x):
E = 1/2 mv² + 1/2 kx²
Therefore, the total energy is constant:
1/2 mv² + 1/2 kx² = 1/2 kA².
SIGNIFICANCE OF ENERGY CONSERVATION
Predicting Behavior:
Applying the energy conservation principle helps predict the system's behavior, like its velo
11/07/2025
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