Center of mass

In physics, the center of mass is a special point in an object or group of objects where the whole mass can be thought of as being concentrated. This point helps us understand how objects move. If you apply a force to the center of mass of a solid object, it will move in a straight line without spinning. This makes many calculations in physics much simpler.

For a single solid object with the same density throughout, the center of mass is at its center. However, this point might not always be inside the object. For example, the center of mass of a horseshoe is actually in the open space above it, not inside the metal.

The center of mass is very important in studying how things move and interact in space. For example, when looking at the planets in our Solar System, scientists treat each planet as if all its mass is at its center of mass to understand their orbits better. This idea helps explain how objects move and stay balanced in space.

History

The idea of the center of mass has very old roots. Ancient Greek thinkers like Archimedes of Syracuse studied it around the 3rd century BCE. Archimedes, a great mathematician, physicist, and engineer, showed how the balance point of objects works, especially in floating bodies. Many other scientists during the Renaissance and Early Modern times also helped develop this idea further.

Definition

The center of mass is a special point in an object or group of objects. It’s the point where the whole mass of the object seems to be concentrated. Imagine balancing a seesaw perfectly level; the center of mass is right where the seesaw balances.

For simple objects like a solid block, the center of mass is right in the middle. For more complex shapes, it can be found by calculating where the weighted average of all the mass points meets. This makes it easier to study how objects move and react to forces.

Center of gravity

Main article: Centers of gravity in non-uniform fields

The center of gravity is the point where the effects of gravity balance out for an object. When gravity is the same everywhere, like on Earth, the center of gravity is the same as the center of mass. But for objects moving in space, gravity changes depending on where you are, so the center of gravity can shift.

The center of mass is a fixed point for a solid object, but the center of gravity can change based on the object's position in a changing gravity field. In most everyday cases, we treat the center of gravity and center of mass as the same because the differences are tiny. Using the center of mass makes it easier to understand how objects move under gravity. When we choose the center of mass as our reference point, the object won't spin because of gravity — it acts like all the weight is at that single point.

Linear and angular momentum

When we look at a group of objects, we can make things easier by thinking about their center of mass. This special point helps us understand how the objects move together.

If we choose the center of mass as our reference point, the math becomes much simpler. The total momentum of all the objects can be found by looking at how the center of mass moves. This idea helps scientists study many different kinds of systems, from magnets to chemical reactions, because it works for many types of forces inside the system.

Determination

See also: Centroid § Determination

We can find an object's center of mass by using gravity. The center of mass is the same as the center of gravity close to Earth's surface.

If an object has a symmetrical shape and the same density all through it, its center of mass will lie on its axis of symmetry. For example, a cylinder's center of mass is along its central axis, and a sphere's center of mass is at its center.

In two dimensions

One way to find the center of mass is to hang the object from two different points and drop a string (called a plumb line) from each point. Where the two strings cross gives the center of mass.

Sometimes shapes are too complicated to calculate directly. In those cases, we can break the shape into simpler parts, find the center of mass for each part, and then find the average for the whole shape. This even works for shapes with holes, by treating the hole as a part with negative mass.

In three dimensions

To find the center of mass in three dimensions, we can support the object at three points and measure the forces holding it up. By measuring these forces at two different heights, we can draw two lines. Where these lines cross gives us the center of mass.

Applications

Engineering designs

Automotive applications

Engineers try to design a sports car so that its center of mass is low. This helps the car turn better and stay on the road even when making sharp turns.

The U.S. military Humvee has a low shape on purpose. This lets it tilt more without tipping over, because its low center of mass stays above its wheels even when turned at sharp angles.

Aeronautics

Main article: Center of gravity of an aircraft

The center of mass is very important for an aircraft. It helps keep the plane stable while flying. If the center of mass is too far forward, the plane may not turn easily. If it is too far back, the plane may be hard to control. For helicopters when hovering, the center of mass is right below the main rotor. When flying forward, it moves ahead to balance the plane.

Astronomy

Main article: Barycenter

The center of mass is also important in space. It is called the barycenter. This is the point where two objects balance each other. For example, when the Moon orbits the Earth, both actually orbit a point between them, not the center of the Earth. This point is where their masses balance. If the objects are similar in size, like Pluto and Charon, this balance point can be outside both objects.

Rigging and safety

Knowing where the center of mass is when lifting things is very important. If the center of mass is too high, the object may tip over. It is safest when the center of mass is low and well below the points where it is being lifted.

Body motion

Main article: Kinesiology

The center of mass of an adult human body is about 10 cm above the hip, slightly forward of the knee, and behind the hip joint. Understanding this helps scientists study how people move. There are two main ways to find it: using a special board to measure while a person lies down, or using math to calculate based on the body’s parts.

Optimization

The center-of-gravity method is a way to solve certain math problems by using the center of mass of possible answers.