Kinematics

Kinematics

Kinematics is a fun part of physics and geometry. It helps us learn about how things move. We don’t need to worry about what forces are pushing or pulling them. Imagine 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 uses many ways to describe positions, such as straight lines like in Cartesian coordinates or curves like in polar coordinates.

This science is very useful for solving real-world problems. It helps 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 looks at how things move. It does this without studying the forces that cause the movement.

Kinematics helps us understand the paths and speeds of objects. These objects can be single points or whole systems.

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

Etymology

The word "kinematic" comes from an old French term made by A.M. Ampère. He created it from a Greek word that means "movement" or "motion." This idea of movement is also linked to the French word for movies, "cinéma," because both words share the same ancient Greek root 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 makes it easier to 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 simpler equations for the object’s velocity and acceleration.

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 thinking of each part as having its own coordinate system, or reference frame. By seeing 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 motion, every part of the object follows the same path as the center of the object. This means the speed and acceleration of each part are the same as the speed and acceleration of the center.

The position of any part in the object can be found by adding the object's movement to its position inside the object. This helps us learn how objects move when they slide without spinning.

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 turn around one fixed point. This helps us understand how things move in circles.

When something turns, we can describe where it is, how fast it moves, and how quickly its speed changes. We call these angular position, angular velocity, and angular acceleration. These ideas are like 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 useful for figuring out how objects move, like the center 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 thinking about the forces that cause the movement. For example, it can describe the paths of connected parts in a machine. 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. There are two main types of these rules. 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.