Euler’s Laws of Motion

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Fifty years after Isaac Newton published his laws of motion in the Principia, Swiss mathematician and physicist Leonhard Euler added to them with his own laws of motion. Euler’s laws take Newton’s laws, which apply only to a singular point of mass, and extend them to an entire rigid body and to rotational motion.

PRINCIPAL TERMS

• angular momentum: the momentum of a rotating object, equal to the object’s moment of inertia (its rotational inertia) times its angular velocity; also called rotational momentum.
• center of mass: the point in an object or system around which the mass of said object or system is evenly distributed.
• fixed reference frame: a frame of reference that is fixed to the environment and not to the subject being observed; sometimes specified as "Earth-fixed" or "space-fixed."
• linear momentum: an object’s mass times its velocity; often called simply "momentum," as the basic concept is defined in terms of an object moving in a straight line.
• Newton’s laws of motion: three laws devised by physicist and mathematician Isaac Newton to describe the motion of objects in relation to the forces acting on them.
• rigid body: an idealization of a solid object that assumes that it cannot be deformed by the forces acting on it.
• torque: the tendency of a force to cause an object to rotate, defined mathematically as the rate of change of the object’s angular momentum; also called moment of force.

Extending Newton’s Laws of Motion

In 1686, English physicist and mathematician Isaac Newton (1643–1727) published his Philosophiae Naturalis Principia Mathematica. In it, he laid out three physical laws that govern interactions between objects and the forces acting on them. These laws are now known as Newton’s laws of motion. The first law states that an object at rest tends to stay at rest, and an object in motion stays in motion, unless acted on by an outside force. The second law states that the net force applied to an object is equal to the resulting change in the object’s momentum per unit time—that is, its mass times its acceleration. The third law states that every action produces an equal and opposite reaction. These laws form the foundation of classical mechanics and, by extension, of all of modern physics.

Newton’s laws are generalizations, devised with reference to a singular point of mass that takes up zero hypothetical space. Fifty years after they were published, Swiss physicist Leonhard Euler (1707–83) extended them by applying them to continuous bodies made up of these Newtonian point masses.

Euler’s First Law of Motion

Mathematically, Newton’s second law is stated as

F = ma

where F is the net force applied to an object (specifically, a point mass), m is its mass, and a is its acceleration. Euler’s first law extends this law to apply to an entire rigid body. A rigid body is a solid object that does not bend, twist, compress, or otherwise deform when acted on by an outside force. When assuming a rigid body, Euler’s first law says, the total force acting on the body is equal to the sum of the forces acting on each individual particle in the body.

Euler’s first law depends on the concept of center of mass. A point mass located at a body’s center of mass and having the same mass as the body will follow the same trajectory as said body when acted on by the same force. For the human body, for example, the center of mass is typically located beneath the belly button. In practical terms, this means that Newton’s first law (F = ma) can be rewritten for a rigid body as

∑F = ma_cm

where ∑F indicates the sum of all external forces acting on all particles of the body (as the symbol ∑ represents summation) and a_cm is the acceleration of the body’s center of mass.

Linear and Angular Momentum

While the term "momentum" is usually used to mean linear momentum, there are in fact two types of momentum. Linear momentum is the momentum of an object moving in a straight line. For a point mass, it is defined as

p = mv

where p is momentum, m is mass, and v is velocity.

In contrast, angular momentum is the momentum of an object that is rotating or traveling in a circle.

Angular momentum (L) is defined as

L = Iω

where I is the moment of inertia and ω is the angular velocity. Moment of inertia, or rotational inertia, is the inertia of a rotating body. It represents an object’s tendency to resist angular acceleration, just as an object tends to resist linear acceleration. Moment of inertia can be further broken down in terms of mass (m) and the distance of the object from the point of rotation (r), that is, the radius of the circular path the object is traveling:

I = r²m

Angular velocity, meanwhile, can be written in terms of the radius of the path of circular motion (r) and the object’s tangential linear velocity (v) at a given instant:

ω = v/r

Thus, in terms of radius, mass, and linear velocity, the equation for angular momentum is as follows:

L = (r²m) (v/r) = rmv

These equations apply to any type of circular motion. They work for spinning motion about an internal axis, such as the rotation of a planet. They also work for motion in a circular path around an external axis, such as the orbit of a planet around the sun.

Euler’s Second Law of Motion

Euler’s second law of motion extends Newton’s laws to apply to rigid bodies in circular motion. It states that for an object rotating about a given point, whether that point is an external axis in a fixed reference frame or the object’s own center of mass, the rate of change of angular momentum is equal to the sum of all external torques, or moments of force, acting about that point. Solving this equation involves differential calculus, specifically finding the derivative of L with respect to time (t). However, if the object is only moving in two dimensions (e.g., a flat, spinning disk), the equation can be rewritten in terms of the center of mass:

∑M = r_cm × a_cmm + Iα

Here, M is moment of force, r_cm is the distance of the object’s center of mass from the axis of rotation, a_cm is the tangential acceleration of the center of mass relative to the axis, and α is the object’s angular acceleration. As before, m is the mass of the object, and I is its moment of inertia.

Sample Problem

A child, Sarah, is riding the merry-go-round at a local playground. The merry-go-round is 3 meters (m) across, and Sarah, who has a mass of 30 kilograms (kg), is sitting on the outside edge. If the merry-go-round completes one full rotation every two seconds (s), calculate Sarah’s angular momentum (L) in units of newton-meter-seconds (n·m·s). Note that one newton-meter-second is equal to one kilogram–square meter per second (kg·m²/s).

Answer:

Because Sarah is so small relative to the circular path of her motion, she can be treated as a point mass, so the simpler equations for circular motion can be used:

L = Iω = r²mω

Her mass is given as 30 kg. She is sitting on the outside edge of the merry-go-round, so the circular path of her motion is the same as the circle formed by the ride. The radius of the ride is simply half the diameter:

r = d/2

r = 3 m / 2

r = 1.5 m

Angular velocity, which measures revolutions per second, is given in units of radians per second (rad/s). One full circle is made up of 2π radians. If Sarah completes one full circle every two seconds, calculate her angular velocity in rad/s:

ω = 2π/2 s

ω = π/s

Plug these values into the equation:

L = r²mω

L = (1.5 m)²(30 kg)(π/s)

L = (2.25 m²)(30 kg)(π/s)

L = 212.058 kg·m²/s = 212.0575 n·m·s

Sarah’s angular momentum is approximately 212.058 newton-meter-seconds.

What Euler’s Laws Mean

Euler’s laws can be more difficult to calculate than the equations derived from Newton’s laws. In particular, applying Euler’s second law in three dimensions requires differential calculus. However, the fundamental idea behind Euler’s math is easy to understand. Newton’s laws assume a single, infinitely small point mass. Euler’s laws assume rigid bodies made up of a continuous collection of these point masses all stuck together. This is a reasonable, if very basic, description of the actual composition of matter: countless tiny atoms bonded together, themselves consisting of subatomic particles linked by fundamental forces.

Bibliography

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