Angular Forces

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: When an object moves in a circular path, as in a planetary orbit, it is subject to an angular force "pulling" it toward the center of the circle. This is called the centripetal force, and in the case of planetary orbits, it is gravity. Newton’s laws for linear motion also describe circular motion and the forces at play.

PRINCIPAL TERMS

• centrifugal force: a fictitious force that seems to push a body in circular motion away from the axis of rotation; in reality, objects in circular motion are subject to centripetal force.
• centripetal force: a force "toward the center" that, in combination with inertia, generates the curved path of an object in circular motion.
• cosine: a trigonometric function describing the relationship between sides of a right triangle; the cosine of an angle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse.
• inertia: the principle that an object at rest tends to stay at rest and an object in motion tends to stay in motion unless acted on by an outside force.
• perpendicular: being at a right angle relative to a given line or plane, as in the lines of the letter T.
• radian: a nondegree unit of angle measurement, based on the radius of a circle; there are 2π radians (equal to 360 degrees) in one complete circle or revolution.
• sine: a trigonometric function describing the relationship between sides of a right triangle; the sine of an angle is equal to the length of the side opposite the angle divided by the length of hypotenuse.
• vector: a quantity with both direction and magnitude.

Forces of Circular Motion

Whenever an object follows a curved path of motion, whether rotating about an internal axis or revolving around an external axis, an angular force is at play. Angular forces are forces that tend to produce circular motion. These forces act in a straight line, as all forces do. However, the result of their influence is a curved path of motion. This is in contrast to linear forces and linear motion, which follow straight lines.

The force that causes objects to follow a curved path is known as centripetal force. "Centripetal" comes from the Latin words for "center-seeking." Any curved motion can be thought of as tracing the circumference of an imaginary circle. The centripetal force always acts toward the center of this circle.

An object moving in a circular path is constantly caught in a tug-of-war between its inertia—that is, its tendency to move in a straight line at a constant speed—and the centripetal force. "Centripetal" describes a category of force and therefore may refer to a variety of forces, such as gravity (planetary orbits), tension (a ball on a string), or even friction (a car turning around a race track). At any given moment, an object in circular motion will tend to continue traveling in a straight line. The centripetal force acts to change this.

Centripetal versus Centrifugal Force

The centripetal force (F_c) needed for an object to travel a circular path is described by the following equation, where m is the mass of the object in motion, v is its linear velocity, and r is the radius of the circular path it is following:

F_c = mv²/r

Force is measured in newtons (N), an International System of Units (SI) derived unit. In SI base units, one newton is equal to one kilogram-meter per second squared (kg∙m/s²). Be sure to use kilograms (not grams) for the mass when making calculations with this equation.

Sometimes centrifugal force is incorrectly used in place of centripetal force. Centrifugal force is a fictitious or illusory force that seems to push outward from an object’s axis of rotation. Imagine being the passenger in a car as it makes a sharp left turn and feeling pressed against the right-hand door. This "force" is not a real force, however. In the example of the turning car, the "force" being felt is that of the car pressing against the passengers to keep them from following their individual inertia and continuing forward in a straight line. When an object is set in circular motion and then released, such as a rock on a string that is spun around and then let go, it is not a "centrifugal force" that propels it outward. Rather, it is the lack of a centripetal force (the tension of the string) that suddenly allows the rock to travel unimpeded in a straight line.

Force Vectors

Forces are vector quantities, meaning that they have both a magnitude and a direction. Vectors are typically indicated using arrows, with the length of the arrow corresponding to the magnitude of the force.

When two vectors are combined, their relative directions determine how they interact. Two forces of equal magnitude forces acting in opposite directions will cancel each other out. Two forces of equal magnitude acting in the same direction result in a doubled total, or net, force. When adding two forces, the angle formed between them determines their net magnitude and direction. Imagine hitting a baseball up and away from home plate. Ignoring drag, two forces are at play. First, the collision imparts the force of the bat’s swing to the ball, pushing it in a straight line up and away toward some point above the horizon. Second, the force of Earth’s gravity pulls the ball downward. The interaction of these forces gives the ball its curved trajectory.

Uniform Circular Motion

In uniform circular motion, an object follows a perfectly circular path at a constant speed. At any given instant, centripetal force acts perpendicular to the linear momentum of the object in motion, forming a right angle between the two competing vectors. These two vectors can be thought of as two sides of a right triangle within the object’s circular path of motion, with the apex of the triangle corresponding to a point along the circle’s circumference. Therefore, trigonometric functions can be used to describe the relationship a given angle of rotation and the vectors at play in circular motion.

The two most commonly used trigonometric functions are sine (sin) and cosine (cos). The sine of a given angle (θ) in a right triangle is equal to the length of the side opposite that angle divided by the length of the triangle’s hypotenuse (the longest side, opposite the right angle):

sinθ = opposite / hypotenuse

The cosine of the angle is equal to the length of the adjacent side divided by the length of the hypotenuse:

cosθ = adjacent / hypotenuse

The preferred unit of angular measure when dealing with circular motion is the radian (rad). Radians are based on the relationship between the radius of a circle and its circumference; there are 2π radians (corresponding to 360 degrees) in one complete revolution. Because circular motion traces the circumference of a circle, radians are also used to describe the angular distance an object travels—that is, how much of its circular path it completes.

Torque is a measure of rotational force acting on an object. Imagine the spoke of a wheel. A hand grips the wheel at some point along the spoke and applies a force perpendicular to the length of the spoke, causing the wheel to rotate around its central pivot point. The radius (r) is the length between that pivot point and the point where the force is applied. Torque (T) is the product of the radius, the force applied (F), and the sine of the angle between their directions (θ):

T = rFsinθ

Another use of these trigonometric functions is in finding the difference between two vectors. This has various applications, such as determining the change in velocity due to a glancing collision. In such a collision, the initial velocity vector and the final (post-collision) velocity vector form an angle at the point of impact, with the initial vector leading toward the impact and the final vector leading away. If the final vector were moved so that its starting point were the same as that of the initial vector, the difference between the two would be equal to a vector leading from the end point of the initial vector to the end point of the final vector, forming the third side of a triangle. The law of cosines says that for a triangle with sides a, b, and c, the length c can be found when the angle C between lengths a and b (opposite side c) is known, according to the following equation:

c² = a² + b² – 2abcosC

This is built into the formula for the cross product, or dot product, of two vectors. (The cross product is the product obtained when two vectors in a three-dimensional space are multiplied, resulting in a third vector perpendicular to both.) That is, the cross product of vectors X and Y is equal to the value 2|X||Y|cosθ. Thus, if the angle (θ) between the initial velocity vector (vi) and the final velocity vector (vf) were known, the length of the difference vector (vd) would be calculated as follows:

|vd|² = (vf − vi)²

|vd|² = (vf − vi)(vf − vi)

|vd|² = (vf × vf) – (vf × vi) – (vf × vi) + (vi × vi)

|vd|² = |vf|² + |vi|² − 2|vf||vi|cosθ

Of course, objects do not always travel with uniform speed. Once an object in circular motion begins to speed up or slow down, the equations above no longer work. As long as the object continues to follow a circular path, the net force acting on the object will always equal the centripetal force, but its magnitude will vary depending on the acceleration of the object (remember, force equals mass times acceleration).

Sample Problem

A car is driving around a circular racetrack one kilometer (km) in circumference. It moves at a constant linear velocity of thirty meters per second (m/s) and has a mass of two thousand kilograms (kg). What is the magnitude of the centripetal force acting on the car?

Answer:

Use the equation for calculating centripetal force:

F_c = mv²/r

Although the radius (r) of the racetrack is not given, it can be determined from the circumference. The equation for finding the circumference of a circle is as follows:

C = 2πr

Rearrange this equation, substitute in the known value of the circumference (C), and solve for r:

C / 2π = r

1 km / 2π = r

r ≈ 0.16 km

Convert the radius from kilometers to meters:

0.16 km × 1,000 m/km = 160 m

Using this value for the radius, as well as the given values for linear velocity (v) and mass (m), calculate the centripetal force (F_c):

F_c = mv²/r

F_c = (2,000 kg)(30 m/s)² / 160 m

F_c = (2,000 kg)(900 m²/s²) / 160 m

F_c = 11,250 kg∙m/s² = 11,250 N

The car is subject to a centripetal force of approximately 11,250 newtons.

Circular Motion in Everyday Life

It is not difficult to find examples of both rotation and revolution in everyday situations. Understanding torque and rotational motion is a vital part of engineering automobiles so that the wheels are given enough force to roll the car forward. "Spin" on an object moving through the air dramatically affects its aerodynamics and trajectory. In tennis, topspin is a vital technique that allows a player to make the ball drop much more sharply than it would under the influence of gravity alone. Although these situations may seem more complicated than the more familiar linear motion of, for instance, billiard balls bouncing around a pool table, it is important to remember that the physical principles underpinning linear and circular motion are the same.

Bibliography

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