Equations of motion determine the speed, displacement, or acceleration of a moving object under a constant linear force. They relate Newton’s second law of motion mathematically and physically, with five variables: displacement, initial and final velocity, acceleration, and time interval. The equations differ for objects moving in a circular path or in a pendulum configuration, with variations accounting for different configurations and assumptions.
The equations of motion are used to determine the speed, displacement, or acceleration of a constantly moving object. Most applications of the equations of motion are used to express how an object moves under the influence of a constant linear force. Variations of the basic equation are used to account for objects moving in a circular path or in a pendulum configuration.
An equation of motion, also called a differential equation of motion, relates Newton’s second law of motion mathematically and physically. The second law of motion, according to Newton, states that a mass under the influence of a force will accelerate in the same direction as the force. Strength and magnitude are directly proportional, and strength and mass are inversely proportional.
The standard equations of motion involve five variables. A variable is for the start and end position of the object, also known as its displacement. Two variables represent the initial and final velocity measurements, known respectively as the velocity change. The fourth variable describes acceleration. The fifth variable represents the time interval.
The classic equation for solving for the linear acceleration of an object is written as the change in velocity divided by the change over time. The law of motion equation is typically set up using three kinetic variables: velocity, displacement, and acceleration. Acceleration can be solved using velocity and displacement as long as the second law of motion applies to the problem.
When an object is in constant acceleration along a rotational path, the equations of motion are different. In this situation, the classical equation for the circular acceleration of an object is written using initial and angular velocities, angular displacement, and angular acceleration.
A more complicated application of the equations of motion is the pendulum equation of motion. The basic equation is known as the Mathieu equation. It is expressed using the constant of gravity for acceleration, pendulum length, and angular displacement.
There are several assumptions that must be satisfied to use such an equation for a problem involving a pendulum configuration. The first assumption is that the rod connecting the mass to the axis point is weightless and remains stretched. The second hypothesis is that movement is limited to two directions, forward and backward. The third assumption is that the energy lost to air resistance or friction is negligible. Variations of the basic equation are used to account for infinitesimal swings, compound pendulums, and other configurations.
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