Center of Mass and Linear Mass Density

The center of mass, also sometimes called the center of gravity, is typically what we refer to as the geometric position in an object defined by: the mean position of every section of the object or system, weighted by mass. In other words, this is a place where the object is balanced in our gravitational field.

Below you can see an example of finding the center of mass in the x direction of a system of masses:

Image from LibreTexts

For a system of masses:

Calculus definition:

Another way to format the above formula is with linear mass density:

Linear mass density is typically a constant for something that is uniform, so it can be found with an equation like:

However, since AP loves to make us do calculus, we will sometimes see non-uniform objects! This means that the linear mass density would be a function. 

Let's try to calculate the center of mass of a uniform rod!

We can begin with one of the formulas we discussed above and place bounds on it:

Since we know that the rod is uniform, we can take the linear mass density out of the integral because it is a constant. 

As you can see, the lambdas cancel out! Now we can evaluate the integrals. 

Now we can plug in our bounds and simplify. This leads us to:

Hopefully, this answer seems intuitive to you! We'll be seeing problems similar to this when we tackle rotational inertia next unit. 

Motion of the Center of Mass

Here are key things to know about the motion of the center of mass:

• The center of mass (COM) of a system is a point that represents the average position of all the matter in the system.
• To find the motion of the COM, you need to know the position and mass of each component of the system.
• The position of the COM can be found using the formula: COM = (m1r1 + m2r2 + ... + mn*rn) / (m1 + m2 + ... + mn)
• The motion of the COM can be found by taking the derivative of its position with respect to time. This will give the velocity of the COM.
• The acceleration of the COM can be found by taking the derivative of its velocity with respect to time. This will give the acceleration of the COM.
• The motion of the COM is useful for understanding the overall motion of a system, rather than the motion of individual components. This type of question is one of the most frequently seen on the AP exam, and it trips a lot of people up! Make sure to read questions carefully and see if they are asking about the center of mass of the system or of the object. 

Center of Gravity vs Center of Mass

Center of gravity (COG) is a point in an object or system where the gravitational force is considered to act. Center of mass (COM) is the point in an object or system where the total mass is considered to be concentrated.

• In a uniform gravitational field, the COG and COM will have identical positions. This means that for an object that is symmetric about an axis and is homogeneous, the COG and COM will be the same.
• For objects or systems that are not symmetric or homogeneous, the COG and COM may have different positions. For example, when an object or system is in a non-uniform gravitational field, or when an object or system is made of different materials, the COG and COM will have different positions.
• In general terms, for any object or system in space, the COM will always be the same regardless of the gravitational field but the COG will be affected by the gravitational field.
• For example, if an object is in orbit around a planet, the COM stays in the same position but the COG will be affected by the gravitational pull of the planet, and it will move as the object orbits.

Practice Questions

Even though AP Physics 1 is not calculus-based, we can practice applications of the center of mass with FRQs from that test too!

Taken from College Board

Answer:

The trick to this question is realizing that it is asking for the center of mass of the system. So the speed of it should only change when momentum isn't conserved, meaning when there is impulse!

Answer:

Same focus as before, realize it is the center of mass of the system! Think of how you were searching for the x coordinate of the center of mass, you can apply the same strategy for velocity.