Kinematics

Kinematics is a fascinating part of physics and geometry. It helps us understand how things move without worrying about what forces are pushing or pulling them. Think of it like watching a ball roll across the ground or a wheel turning — kinematics tells us where the ball or wheel is, how fast it’s going, and how its path changes over time.

In geometry, kinematics looks at how shapes and positions change as time passes. It studies things like how far apart points are or how angles shift, all in relation to a fixed point or direction, called a frame of reference. Kinematics works with many different ways to describe positions, such as using straight lines like in Cartesian coordinates or curves like in polar coordinates.

This area of science is very useful for solving real-world problems, especially when things are connected and must move in certain ways, like the links in a machine or the wheels on a car. By understanding kinematics, scientists and engineers can design everything from simple tools to complex robots!

Overview

Kinematics is a part of physics and mathematics that studies how things move without looking at the forces that make them move. It helps us understand the paths and speeds of objects, whether they are single points or whole systems.

People use kinematics in many areas, like studying stars in space or designing robot arms. It is also important in engineering and biology to describe how parts of machines or bodies, like our skeleton, move. Kinematics helps simplify complex movements and is even used in advanced theories like relativity.

Etymology

The word "kinematic" comes from an old French term created by A.M. Ampère. He built it from a Greek word meaning "movement" or "motion." This idea of movement connects to the French word for movies, "cinéma," because both come from the same ancient Greek root word for movement.

Kinematics of a particle trajectory in a non-rotating frame of reference

Particle kinematics is the study of how particles move through space. The position of a particle is described by a vector that shows where it is from a starting point, called the origin. For example, if you imagine your home as the origin, a tower 50 meters south and 50 meters tall would have a position vector showing its location.

The motion of a particle can be described using three main ideas: position, velocity, and acceleration. Position tells us where the particle is. Velocity tells us how fast the particle is moving and in what direction. Acceleration tells us how the velocity is changing — either in speed, direction, or both. These ideas help us understand and predict the path a particle will take as it moves.

Particle trajectories in cylindrical-polar coordinates

See also: Generalized coordinates, Curvilinear coordinates, Orthogonal coordinates, and Frenet-Serret formulas

It’s helpful to describe the path of a moving object using polar coordinates on the X-Y plane. This way, we can easily find its speed and how it changes direction.

When an object moves along the surface of a circular cylinder, we can align the Z-axis with the cylinder’s center. The angle around this axis helps us track the object’s position. By using special directions called radial and tangential unit vectors, we can write equations for the object’s velocity and acceleration in a simpler way.

For objects moving in circles, their acceleration has two parts: one toward the center (centripetal acceleration) and one that changes how fast they spin (Coriolis acceleration).

Point trajectories in a body moving in the plane

The movement of parts in a machine can be studied by imagining each part has its own coordinate system, or reference frame. By looking at how these reference frames move in relation to each other, we can understand the paths that different points on the parts follow.

Geometry helps us describe these movements. When we move shapes in a plane without changing their size or angles, we use special tools called rigid transformations. These include turning (rotating) and sliding (translating) objects. In two dimensions, we can use special math tools, called matrices, to describe these movements exactly. This helps us predict where every point on a moving part will be at any time.

Pure translation

If a solid object moves without turning, it is called pure translation. In this type of motion, every point in the object moves along the same path as the object's center. This means that the speed and acceleration of each point are the same as the speed and acceleration of the center of the object.

The position of any point in the object can be described by adding the object's movement to its position within the object. This helps us understand how objects move when they slide without rotating.

Main article: reference frame

Rotation of a body around a fixed axis

Main article: Rotation around a fixed axis

Objects like a playground merry-go-round or a fan can be thought of as solid objects turning around one fixed point. This helps us understand how things move in circles.

When something turns, we can describe its position, speed, and how fast its speed changes. We call these angular position, angular velocity, and angular acceleration. These ideas are similar to how we describe straight-line motion, but they apply to spinning or rotating objects. For example, we can calculate how fast a fan blade moves or how quickly it speeds up or slows down.

Point trajectories in body moving in three dimensions

Important ideas in kinematics help us understand how points move in three-dimensional space. This is very useful for figuring out the motion of objects like the center of mass of a body. We can use special formulas to describe this motion, either through Newton's second law or Lagrange's equations.

Kinematics looks at how objects move without worrying about the forces that cause the movement. For example, it can describe the paths of connected machine parts. The velocity and acceleration of points in a moving body follow specific patterns that can be calculated using matrices and vectors. These calculations help scientists and engineers predict how objects will move in space.

Kinematic constraints

Kinematic constraints are rules that control how parts of a machine can move. These constraints come in two main types. The first type includes joints like hinges and sliders, which decide how the machine is built. The second type includes rules about speed, like how ice-skates slide on flat ice.

A kinematic coupling stops all movement in six directions. When something rolls without slipping, like a wheel on the ground, its speed depends on how fast it turns. Cords that can't stretch connect parts of a machine, like in a pendulum.

Kinematic pairs are connections between parts of a machine. Lower pairs include joints like hinges and sliders, while higher pairs involve surfaces touching each other, like gears. Kinematic chains are links connected by these pairs, and they can have different numbers of moving parts and ways to connect them. For example, a four-bar linkage has four links and four joints.