## Table of Contents

- Center of Mass Formula
- Velocity of Center of Mass
- Velocity of Center of Mass Equation
- Acceleration of Center of Mass
- Mass Times Velocity
- Lesson Summary

- Center of Mass Formula
- Velocity of Center of Mass
- Velocity of Center of Mass Equation
- Acceleration of Center of Mass
- Mass Times Velocity
- Lesson Summary

The **center of mass** is the average position of all the particles within an object or system of objects. For simple objects with constant densities, finding the center of mass can be intuitive. A square's center of mass is in the geometric center of the square, also known as the centroid, because that is the average position of all the square's mass. However, sometimes more complex shapes or multiple objects in a system make it difficult to know where the center of mass is.

The center of mass formula for a system of objects (in this case, three objects) can be found using: {eq}X = \frac{m1*x1*m2*x2*m3*x3}{m1+m2+m3} {/eq}

- X = the position of the center of mass
- m = the mass of an object (kg)
- x = the position of an object

Note: the position unit can vary as long as it remains consistent throughout the work.

**Velocity** is the change in an object's position over time. Since the center of mass moves along with an object or system of objects in motion, the overall movement can be described in terms of the velocity of center of mass.

Staying with the three-object-system that was used in the previous formula, the velocity of center of mass equation is the sum of the momentum of the three objects in the system, divided by the total mass:

{eq}V_{cm} = \frac{(m_1v_1) + (m_2v_2) + (m_3v_3)}{m_1+m_2+m_3} {/eq}

- V = velocity of the center of mass (m/s)
- m = the mass of an object (kg)
- v = velocity of an object (m/s)

Let's say there are three objects. One with a mass of 2 kg, one with a mass of 5 kg, and one with a mass of 8 kg. The first object has a velocity of 1 m/s, the second has a velocity of 4 m/s, and the last has a velocity of 2 m/s. The velocity of the center of mass can be found using the formula above.

{eq}V_{cm} = \frac{(2 kg*1 m/s) + (5 kg*4 m/s) + (8 kg*2 m/s)}{2 kg+5 kg+8 kg} {/eq}

Simplified, this equals:

{eq}V_{cm} = \frac{38 kg*m/s}{15 m/s} {/eq}

Which equals approximately 2.5 m/s for the velocity for the center of mass.

**Acceleration** occurs when velocity changes in relation to time. Acceleration of an object can be found using the formula:

F = m * a

F = force (N)

m = mass (kg)

a = m/s^2

The acceleration of center of mass can be found using the formula:

{eq}a_{cm} = \frac{m_1a_1 + m_2a_2 + ... + m_na_n}{m_1 + m_2 + ... m_n} = \frac{m_1a_1 + m_2a_2 + m_na_n}{M} {/eq}

- The acceleration of the center of mass is {eq}a_{cm} {/eq}

- The acceleration of particle 1, etc. is {eq}a_1 {/eq}

- The mass of particle 1, etc. is {eq}m_1 {/eq}

- The total mass is M

Multiplying both sides by M gives:

{eq}Ma_{cm} = m_1a_1 + m_2a_2 + ... + m_na_n {/eq}

- {eq}m_1a_1 {/eq} is the net force of particle 1

- {eq}m_2a_2 {/eq} is the net force of particle 2

- {eq}m_na_n {/eq} is the net force of n particles

Example:

In this example there are three objects. The mass of object 1 is 5 kg. The mass of object 2 is 2 kg. The mass of object 3 is 8 kg. The acceleration for object 1 is 2 m/s. The acceleration for object 2 is 1 m/s The acceleration for object 3 is 5 m/s.

Start with the formula {eq}a_{cm} = \frac{m_1a_1 + m_2a_2 + ... + m_na_n}{m_1 + m_2 + ... m_n} = \frac{m_1a_1 + m_2a_2 + m_na_n}{M} {/eq}

Plug in the known values {eq}a_{cm} = \frac{5*2 + 2*1 + 3*5}{5 + 2 + 3} {/eq}

Simplify {eq}a_{cm} = \frac{10 + 2 + 15}{10} {/eq}

{eq}a_{cm} = \frac{27}{10} {/eq} or 2.7 m/s

**Momentum** is the mass of an object in motion. It is the product of the mass of an object and the object's velocity.

The basic formula for momentum is:

p=m*v

momentum = mass times velocity

The formula for momentum appears in the calculation of the velocity of the center of mass. The original formula is:

{eq}V_{cm} = \frac{(m_1v_1) + (m_2v_2) + (m_3v_3)}{m_1+m_2+m_3} {/eq}

Rearranged and multiplied by M, the formula becomes:

{eq}Mv_{cm} = m_1v_1 + m_2v_2 + ... + m_nv_n {/eq}

Example:

A person is riding a bike at a constant velocity of 15 m/s. The mass of the person plus the mass of the bike is 125 kg. What is the momentum of the bike and the rider?

p=mv

p=(125)(15)

p=1,875 kg*m/s

Example:

A ball is rolling down a hill. The mass of the ball is 0.5 kg and the ball is rolling at a velocity of 20 m/s. What is the momentum of the soccer ball?

p=mv

p=(0.5)(20)

p=10 kg*m/s

Derivative for the Formula of the Center of Mass:

- Start with the center of mass formula:

{eq}x_{cm} = \frac{m_1x_1 + m_2x_2 + ... + m_nx_n}{m_1 + m_2 + ... + m_n} {/eq}

- Take the derivative of both sides:

{eq}\frac{d}{dt}(x_{cm}) = \frac{d}{dt}(\frac{m_1x_1 + m_2x_2 + ... + m_nx_n}{m_1 + m_2 + ... + m_nx_n}) = \frac{m_1\frac{d}{dt}(x_1) + m_2\frac{d}{dt}(x_2) + ... + m_n\frac{d}{dt}(x_n)}{m_1 + m_2 + ... + m_n} {/eq}

- Therefore:

{eq}v_{cm} = \frac{m_1v_1 + m_2v_2 + ... + m_nv_n}{m_1 + m_2 + ... + m_n} {/eq}

Finding the **center of mass** for a simple object with constant density can be simple. The center of mass is the average position of all the particles within an object or system of objects. Finding the center of mass for a simple object, like a square, is relatively simple, the center of mass is the geometric center, or centroid, of the square. For more complex and irregular shapes, the center of mass formula can be used. The center of mass formula is:

{eq}X = \frac{m1*x1*m2*x2*m3*x3}{m1+m2+m3} {/eq}

- X = the position of the center of mass
- m = the mass of an object (kg)
- x = the position of an object

**Velocity** is an object's change in position over time. This is calculated by taking the sum of the momentums and dividing it by the total mass. The** acceleration** for the center of mass is its change in velocity in relation to time. It's calculated by the total external force divided by the total mass. The mass of an object in motion is it's **momentum**. Momentum can also be defined as the product of an object's mass and its velocity.

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Frequently Asked Questions

The center of mass is simple the geometric center of the shape. For more complex shapes, the center of mass formula must be used.

The formula for the center of mass is the sum of the product of the mass and position of each particle in the system. Take that and divide it by the sum of the mass of each particle.

To find the center of mass of a system of particles, you must know the mass and position of each particle. Add up the sum of the product of the mass and position of each particle and divide it by the sum of the mass of each particle.

Yes, the velocity of the center of mass is constant. The exception would be if it is acted on by an outside force.

The center of mass for a simple object like a square, is its geometric center (centroid). To find the center of a more complex object, the center of mass formula is needed. The formula for a system of particles is the sum of the product of the mass and position of each particle, divided by the total mass.

The velocity of the center of mass for a group of objects is the sum of the product of each object's mass and velocity, divided by the total mass. The momentum of an object is the product of its mass and velocity.

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