Rotational Motion

Rotational Motion (Physics): What is it & Why it Matters

Updated December 28, 2020

By Kevin Beck

Perhaps you think of your movements in the world, and the motion of objects in general, in terms of a series of mostly straight lines: You walk in straight lines or curved paths to get from place to place, and rain and other things fall from the sky; much of the world’s critical geometry in architecture, infrastructure and elsewhere is predicated on angles and carefully arranged lines. At a glance, life may seem far more rich in linear (or translational) motion than in angular (or rotational) motion.

As with a lot of human perceptions, this one, to the extent each persons experiences it, is hugely misleading. Thanks to how your senses are structures to interpret the world, it is natural for you to navigate that world in terms of forward and back and right and left and up and down. But were it not for rotational motion – that is, motion about a fixed axis – there would be no universe or at least not one hospitable or recognizable to physics buffs.What is thermodynamics & law of Thermodynamics?

Okay, so things spin around as well as shift about generally. What of it? Well, the big takeaways about rotational motion are that: 1) It has mathematical analogs in the world of linear or translational motion that make studying either one in the context of the other extremely useful, as it shows how physics itself is “set up”; and 2) the things that set rotational motion apart are very important to learn.

What Is Rotational Motion?

Rotational motion refers to anything spinning or moving in a circular path. It is also called angular motion or circular motion. The motion may be uniform (i.e., the velocity v doesn’t change) or non-uniform, but it has to be circular. Brought to you by Sciencing

The revolution of the Earth and other planets around the sun may be treated as circular for simplicity, but planetary orbits are actually elliptical (slightly oval) and therefore not an example of rotational motion.https://eblog.mhaagj.org

An object can be rotating while also experiencing linear motion; just consider a football spinning like a top as it also arcs through the air, or a wheel rolling down the street. Scientists consider these kinds of motion separately because separate equations (but again, tightly analogous) are required to interpret and explain them. It’s actually useful to have a special set of measurements and calculations to describe rotational motion of those objects as opposed to their translational or linear motion, because you often get a brief refresher in things like geometry and trigonometry, subjects it is always good for the science-minded to have a firm handle on.

Why Studying Rotational Motion Matters

While the ultimate non-acknowledgment of rotational motion might be “Flat Earthism,” it is actually pretty easy to miss even when you’re looking, perhaps because many people’s minds are trained to equate “circular motion” with “circle.” Even the tiniest slice of the path of an object in rotational motion around a very distant axis – which would look like a straight line at a glance – represents circular motion.

Such motion is all around us, with examples including rolling balls and wheels, merry-go-rounds, spinning planets and elegantly twirling ice-skaters. Examples of motions that may not seem like rotational motion, but in fact are, include see-saws, opening doors and the turn of a wrench. As noted above, because in these cases the angles of rotation that are involved are often small, it’s easy to not filter this in your mind as angular motion.

Think for a moment about the motion of a cyclist with respect to the “fixed” ground. While it’s obvious that the wheels of the bike are moving in a circle, consider what it means for the cyclist’s feet to be fixed to the pedals while the hips remain stationary atop the seat.

The “levers” in between are executing a form of complex rotational motion, with the knees and ankles tracing out invisible circles with different radii. Meanwhile, the whole package might be moving at 60 km/hr through the Alps during the Tour de France.

Newton’s Laws of Motion

Hundreds of years ago, Isaac Newton, perhaps the most high-impact math and physics innovator in history, produced three laws of motion that he based largely on the work of Galileo. Since you are studying motion formally, you might as well be familiar with the “ground rules” governing all motion and who discovered them.

Newton’s first law, the law of inertia, states that an object moving with constant velocity continues to do so unless disturbed by an external force. Newton’s second law proposes that if a net force F acts on a mass m, it will accelerate (change the velocity of) that mass in some way: F = ma. Newton’s third law states that for every force F there exists a force –F, equal in magnitude but opposite in direction, so that the sum of the forces in nature is zero. Rotational Motion vs. Translational Motion

In physics, any quantity that can be described in linear terms can also be described in angular terms. The most important of these are:

Displacement. Usually, kinematics problems involve two linear dimensions to specify position, x and y. Rotational motion involves a particle at a distance r from the axis of rotation, with an angle specified in reference to a zero point if needed.

Velocity. Instead of velocity v in m/s, rotational motion has angular velocity ω (the Greek letter omega) in radians per second (rad/s). Importantly, however, a particle moving with constant ω also has a tangential velocity vt in a direction perpendicular to r. Even if constant in magnitude, vt is always changing because the direction of its vector continually changes. Its value is found simply from vt = ωr.

Acceleration. Angular acceleration, written α (The Greek letter alpha), is often zero in basic rotational motion problems because ω is usually held constant. But because vt, as noted above, is always changing, there exists a centripetal acceleration ac directed inward toward the rotation axis and with a magnitude of

Force. Forces that act about an axis of rotation, or “twisting” (torsional) forces, are called torques, and are a product of the force F and the distance of its action from the axis of rotation (i.e., the length of the lever arm):

τ=F×r

Note that the units of torque are Newton-meters, and the “×”here signifies a vector cross product, indicating that the direction of τ is perpendicular to the plane formed by F and r.

Mass. While mass, m, factors into rotational problems, it is usually incorporated into a special quantity called the moment of inertia (or second moment of area) I. You’ll learn more about this actor, along with the more fundamental quantity angular momentum L, soon.

Radians and Degrees

Because rotational motion involves studying circular paths, rather than using meters to describe the angular displacement of an object, physicists use radians or degrees. A radian is convenient because it naturally expresses angles in terms of π, since one complete turn of a circle (360 degrees) equals 2π radians. Commonly encountered angles in physics are 30 degrees (

π/6 rad), 45 degrees (π/4 rad), 60 degrees (π/3 rad) and 90 degrees (π/2 rad).

Axis of Rotation

Being able to identify the axis of rotation is essential in understanding rotational motions and solving associated problems. Sometimes this is straightforward, but consider what happens when a frustrated golfer sends a five-iron twirling high into the air toward a lake.

A single rigid body con rotate in a surprising number of ways: end-over-end (like a gymnast doing 360-degree vertical spins while holding a horizontal bar), along the length (like the drive shaft of a car), or spinning from a central fixed point (like the wheel of that same car).

Typically, the properties of an object’s motion change depending on how it is rotated. Consider a cylinder, half of which is made of lead and the other half of which is hollow. If an axis of rotation were chosen through its long axis, the distribution of mass around this axis would be symmetrical, though not uniform, so you can imagine it spinning smoothly. But what if the axis were chosen through the heavy end? The hollow end? The middle?

Moment of Inertia

As you just learned, spinning the same object around a different axis of rotation, or changing the radius, can make the motion more or less difficult. A natural extension of this concept is that similarly shaped objects with different distributions of mass have different rotational properties.

This is captured by a quantity called the moment of inertia I, which is a measure of how hard it is to change an object’s angular velocity. It is analogous to mass in linear motion in terms of its general effects on rotational motion. As with elements in the periodic table in chemistry, it’s not cheating to look up the formula for I for any object; a handy table is found in the Resources. But for all objects, I is proportional to both mass (m) and the square of the radius (r2).

The biggest role of I in computational physics is that it offers a platform for computing angular momentum L:

L=Iω

Conservation of Angular Momentum

The law of conservation of angular momentum in rotational motion is analogous to the law of conservation of linear momentum and is a critical concept in rotational motion. Torque, for example, is just a name for the rate of change of angular momentum. This law states that the total momentum L in any system of rotating particles or objects never changes.

This explains why an ice skater spins so much faster as she pulls in her arms, and why she spreads them out to slow herself to a strategic stop. Recall that L is proportional to both m and r2 (because I is, and L = Iω). Because L must remain constant, and the value of m (the skater’s mass doesn’t change during the problem, if r increases, then the final angular velocity ω must decrease and conversely.

Centripetal Force

You’ve already learned about centripetal acceleration ac, and that where acceleration is in play, so is force. A force that compels an object follow a curved path is subject to a centripetal force. A classic example: The tension (force per unit length) on a string holding a tether ball is directed toward the center of the pole and makes the ball keep moving around the pole.

This causes centripetal acceleration toward the center of the path. As noted above, even at constant angular velocity, an object has centripetal acceleration because the direction of the linear (tangential) velocity vt is continually changing.

A vector has both a magnitude and a specific direction, but a scalar quantity only has a magnitude.

Vectors vs. Scalars

The key difference between vectors and scalars is that a vector’s magnitude doesn’t entirely describe it; there also needs to be a stated direction.

The direction of a vector can be stated in numerous ways, whether through positive or negative signs in front of it, expressing it in the form of components (scalar values next to the appropriate i, j and k “unit vector,” which correspond to the Cartesian coordinates of x, y and z, respectively), adding an angle with respect to a stated direction (e.g., “60 degrees from the x-axis”) or simply adding some words to describe the direction (e.g., “northwest”).

By contrast, a scalar is just the vector’s magnitude without any additional notation or information provided – for example, speed is a scalar equivalent of the velocity vector. From a mathematical perspective, it’s the absolute value of the vector.

However, many quantities, such as energy, pressure, length, mass, power and temperature are examples of scalars that aren’t just the magnitude of a corresponding vector. You don’t need to know the “direction” of mass, for example, to have a complete picture of it as a physical property.

There are a few counterintuitive facts that you can understand when you know the difference between a scalar and a vector, such as the idea that something could have a constant speed but a continuously changing velocity. Imagine a car driving at a constant speed of 10 km/h but in a circle. Because the direction of a vector is part of its definition, the car’s velocity vector is always changing in this example, despite the fact that the magnitude of the vector (i.e., its speed) is constant.

Examples of Vector Quantities

There are many examples of vectors in physics, but some of the most well-known examples are force, momentum, acceleration and velocity, all of which feature strongly in classical physics. A velocity vector could be displayed as 25 m/s to the east, −8 km/h in the y-direction, v = 5 m/s i + 10 m/s j, or 10 m/s in a direction 50 degrees from the x-axis.

Momentum vectors are another example you can use to see how the magnitude and direction of the vector are displayed in physics. These work just like the velocity vector examples, with 50 kg m/s to the west, −12 km/h in the z direction, p = 12 kg m/s i – 10 kg m/s j – 15 kg m/s k and 100 kg m/s 30 degrees from the x-axis being examples of how they could be displayed. The same basic points go for the display of acceleration vectors, with the only difference being the unit of m/s2 and the commonly-used symbol for the vector, a.

Force is the final one of these examples of vector expressions, and while there are many similarities, using cylindrical coordinates (r, θ, z) instead of Cartesian coordinates can help to show other ways they may be displayed. For example, you might write a force as F = 10 N r + 35 N 𝛉, for a force with components in the radial direction and the azimuthal direction, or describe the force of gravity on a 1-kg object on Earth as 10 N in the –r direction (i.e., towards the center of the planet).

Vector Notation in Diagrams

In diagrams, vectors are displayed using arrows, with the magnitude of the vector represented by the length of the arrow and its direction represented by the direction in which the arrow points. For example, a larger arrow shows that a force is larger (i.e., more newtons or a bigger magnitude) than another force.

For a vector that shows motion, such as the momentum or the velocity vector, the zero vector (i.e., a vector representing no velocity or momentum) is displayed using a single dot.

It’s worth noting that because the length of the arrow represents the magnitude of the vector and its orientation represents the direction of the vector. It’s useful to try to be reasonably accurate when making a vector diagram. It doesn’t have to be perfect, but if the vector a is twice as big as the vector b, the arrow should be roughly twice as long.

Vector Addition and Subtraction

Vector addition and vector subtraction are a bit more complicated than adding and subtracting scalars, but you can pick up the concepts easily. There are two main approaches you can use, and each has potential uses depending on the specific problem you’re tackling.

The first, and the easiest to use when you’ve been given two vectors in component form, is to simply add matching components in the same way you’d add ordinary scalars. For example, you needed to add the two forces F1 = 5 N i + 10 N j and F2 = 6 N i + 15 N j + 10 N k, you would add the i components, then the j components and finally the k components

=(5Ni+10Nj)+(6Ni+15Nj+10Nk)

=(5N+6N)i+(10N+15N)j+(0N+10N)k

=11Ni+25Nj+10Nk

Vector subtraction works in exactly the same way, except you subtract the quantities rather than add them. Vector addition is also commutative, like ordinary addition with real numbers, so a + b = b + a.

You can also perform vector addition using arrow diagrams by laying the vector arrows head to tail and then drawing a new vector arrow for the sum of the vectors connecting the tail of the first arrow with the head of the second.

If you have a simple vector addition with one in the x-direction and another in the y-direction, the diagram forms a right-angled triangle. You can complete the vector addition and determine the resulting vector’s magnitude and direction by “solving” the triangle using trigonometry and Pythagoras’ theorem.

The Dot Product and Cross Product

Multiplying vectors is a bit more complicated than scalar multiplication for real numbers, but the two main forms of multiplication are the dot product and the cross product. The dot product is called the scalar product and is defined as:

u∙v=∣u∣∣v∣cos(θ)

where θ is the angle between the two vectors, and the subscripts 1, 2 and 3 represent the first, second and third component of the vector. The result of the dot product is a scalar.