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The is a crucial concept in physics, representing the average position of mass in a system. It simplifies complex motion analysis by treating entire objects as single points, making it easier to understand trajectories, collisions, and stability.

Calculating the involves weighted averages for discrete systems or integrals for continuous ones. Its motion follows Newton's laws, with velocity and acceleration determined by external forces. This principle is key in analyzing collisions, connected objects, and rotational motion.

Center of Mass

Definition and significance of center of mass

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  • Represents the average position of all mass in a system, as if total mass were concentrated at a single point
    • Depends on within the system (uniform vs non-uniform)
    • Examples: center of a uniform sphere, off-center for an irregularly shaped object
  • Simplifies analysis of a system's motion by treating it as a single point mass
    • Overall motion represented by center of mass trajectory (projectile motion, orbits)
    • External forces considered to act at the center of mass (gravity, friction)
  • Crucial for understanding rotational motion, collisions, and object stability
    • Determines the axis of rotation and (spinning top, gyroscope)
    • Influences energy transfer and momentum conservation during collisions (billiard balls)
    • Affects balance and tipping points (leaning tower, balancing sculptures)

Calculation methods for center of mass

  • Discrete systems: use weighted average of particle positions
    • x-coordinate: [xcm](https://www.fiveableKeyTerm:xcm)=∑i=1nmixi∑i=1nmi[x_{cm}](https://www.fiveableKeyTerm:x_{cm}) = \frac{\sum_{i=1}^{n} m_i x_i}{\sum_{i=1}^{n} m_i}, where mim_i is mass and xix_i is x-position of i-th particle
    • y-coordinate: [ycm](https://www.fiveableKeyTerm:ycm)=∑i=1nmiyi∑i=1nmi[y_{cm}](https://www.fiveableKeyTerm:y_{cm}) = \frac{\sum_{i=1}^{n} m_i y_i}{\sum_{i=1}^{n} m_i}, yiy_i is y-position of i-th particle
    • z-coordinate: [zcm](https://www.fiveableKeyTerm:zcm)=∑i=1nmizi∑i=1nmi[z_{cm}](https://www.fiveableKeyTerm:z_{cm}) = \frac{\sum_{i=1}^{n} m_i z_i}{\sum_{i=1}^{n} m_i}, ziz_i is z-position of i-th particle
  • Continuous systems: use mass density ρ(x,y,z)\rho(x, y, z) and volume integrals
    • x-coordinate: xcm=∫xρ(x,y,z)[dV](https://www.fiveableKeyTerm:dV)∫ρ(x,y,z)dVx_{cm} = \frac{\int x \rho(x, y, z) [dV](https://www.fiveableKeyTerm:dV)}{\int \rho(x, y, z) dV}, dVdV is volume element
    • y-coordinate: ycm=∫yρ(x,y,z)dV∫ρ(x,y,z)dVy_{cm} = \frac{\int y \rho(x, y, z) dV}{\int \rho(x, y, z) dV}
    • z-coordinate: zcm=∫zρ(x,y,z)dV∫ρ(x,y,z)dVz_{cm} = \frac{\int z \rho(x, y, z) dV}{\int \rho(x, y, z) dV}
  • simplifies calculations
    • Lies on axis or plane of symmetry (rod, rectangle, sphere)
    • Reduces dimensions needed for calculation (2D vs 3D)

Motion of system's center of mass

  • Velocity [v⃗cm](https://www.fiveableKeyTerm:v⃗cm)[\vec{v}_{cm}](https://www.fiveableKeyTerm:\vec{v}_{cm}) is time derivative of position [r⃗cm](https://www.fiveableKeyTerm:r⃗cm)[\vec{r}_{cm}](https://www.fiveableKeyTerm:\vec{r}_{cm})
    • v⃗cm=dr⃗cmdt=∑i=1nmiv⃗i∑i=1nmi\vec{v}_{cm} = \frac{d\vec{r}_{cm}}{dt} = \frac{\sum_{i=1}^{n} m_i \vec{v}_i}{\sum_{i=1}^{n} m_i}, v⃗i\vec{v}_i is velocity of i-th particle
    • Represents average velocity of all particles in the system
  • Acceleration [a⃗cm](https://www.fiveableKeyTerm:a⃗cm)[\vec{a}_{cm}](https://www.fiveableKeyTerm:\vec{a}_{cm}) is time derivative of velocity v⃗cm\vec{v}_{cm}
    • a⃗cm=dv⃗cmdt=∑i=1nmia⃗i∑i=1nmi\vec{a}_{cm} = \frac{d\vec{v}_{cm}}{dt} = \frac{\sum_{i=1}^{n} m_i \vec{a}_i}{\sum_{i=1}^{n} m_i}, a⃗i\vec{a}_i is acceleration of i-th particle
    • Represents average acceleration of all particles in the system
  • Follows Newton's second law: ∑F⃗ext=[Ma⃗cm](https://www.fiveableKeyTerm:Ma⃗cm)\sum \vec{F}_{ext} = [M\vec{a}_{cm}](https://www.fiveableKeyTerm:M\vec{a}_{cm})
    • ∑F⃗ext\sum \vec{F}_{ext} is sum of external forces, MM is total mass
    • Internal forces cancel out and do not affect center of mass motion

Applications in collisions and complex motion

  • Constant velocity in absence of external forces
    • Useful for analyzing motion before and after collisions (elastic vs inelastic)
    • Helps determine initial and final velocities of colliding objects
  • during collisions
    • Total momentum of system remains constant
    • Allows calculation of post-collision center of mass velocity
  • Treating connected objects as a single point mass
    • Simplifies analysis of systems with internal forces (pulleys, springs)
    • Center of mass motion determined by external forces only
  • Calculating and rotational motion
    • Center of mass often the point about which torque is calculated
    • Determines the axis of rotation and (pendulum, flywheel)

Fundamental principles and their relation to center of mass

  • govern the behavior of center of mass
    • First law: center of mass remains at rest or in uniform motion unless acted upon by external forces
    • Second law: net determines acceleration of center of mass
    • Third law: internal forces cancel out, not affecting center of mass motion
  • of the system is directly related to center of mass motion
    • Total linear momentum is product of total mass and center of mass velocity
  • of the system is calculated relative to the center of mass
    • Simplifies analysis of rotational motion in many cases
  • of the system is represented by the mass distribution around the center of mass
    • Affects how easily the system's rotational motion can be changed

Key Terms to Review (34)

\vec{a}_{cm}: \vec{a}_{cm}$ is the acceleration of the center of mass of a system of particles or a rigid body. It represents the overall acceleration experienced by the center of mass, which is the point where the entire mass of the system can be considered to be concentrated.
\vec{r}_{cm}: \vec{r}_{cm}$ is the position vector of the center of mass of a system of particles or a rigid body. It represents the average position of all the particles or mass elements that make up the system, weighted by their individual masses.
\vec{v}_{cm}: \vec{v}_{cm}, or the center of mass velocity, is a vector quantity that represents the velocity of the center of mass of a system. The center of mass is the point where the entire mass of the system can be considered to be concentrated, and its velocity describes the overall motion of the system.
∑ \vec{F}_{ext}: The sum of all external forces acting on a system. This term is crucial in understanding the concept of center of mass, as the net external force on a system determines the acceleration of its center of mass.
∑miri/M: ∑miri/M is a mathematical expression that represents the center of mass of a system. It is the weighted average of the positions of the individual masses within the system, where the weights are the masses themselves.
Angular momentum: Angular momentum is a measure of the quantity of rotation of an object and is a vector quantity. It is given by the product of the moment of inertia and angular velocity.
Angular Momentum: Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is the measure of an object's rotational inertia and its tendency to continue rotating around a specific axis. Angular momentum is a vector quantity, meaning it has both magnitude and direction, and it plays a crucial role in understanding the behavior of rotating systems across various topics in physics.
Archimedes: Archimedes was a renowned ancient Greek mathematician, physicist, engineer, and inventor. He is best known for his principles related to buoyancy and the concept of the center of mass, which have important applications in physics and engineering.
Barycenter: The barycenter, or center of mass, is the average position of the mass in a system. It is the point around which the system's mass is evenly distributed, and it is the point that the system's motion can be described as if all the mass were concentrated there.
Center of mass: The center of mass is the point in an object or system where all its mass can be considered to be concentrated for the purpose of analyzing translational motion. It is the weighted average position of all the mass in the system.
Center of Mass: The center of mass is the point at which an object's entire mass can be considered to be concentrated. It is the average position of the mass of an object, and it is the point around which the object's rotation and motion can be analyzed.
Centroid: The centroid is the point at which an object's mass is evenly distributed, or the geometric center of a shape. It is a fundamental concept in the study of center of mass, which is a crucial topic in classical mechanics.
Conservation of Momentum: Conservation of momentum is a fundamental principle in physics that states the total momentum of a closed system remains constant unless an external force acts upon it. This principle applies to various topics in mechanics, including Newton's Third Law, linear momentum, impulse and collisions, types of collisions, center of mass, and rocket propulsion.
DV: dV, or the differential of volume, is a fundamental concept in physics that represents an infinitesimally small change in the volume of an object or system. It is a crucial term in the study of the Center of Mass, as it is used to calculate the total mass and the position of the center of mass of an object or system.
External force: An external force is any force that originates from outside a system and acts on the objects within that system. It can change the motion of the objects by altering their velocity or causing deformation.
Inertia: Inertia is the property of an object that resists changes to its state of motion. It depends solely on the mass of the object.
Inertia: Inertia is the property of an object that resists changes to its state of motion. It is the tendency of an object to remain at rest or in motion unless acted upon by an unbalanced force.
Internal force: Internal force is a force that acts between particles within a system, causing interactions but not changing the overall momentum of the system. These forces are responsible for maintaining the integrity and shape of the system.
Kg·m: kg·m, or kilogram-meter, is a unit of angular momentum or moment of inertia in the International System of Units (SI). It represents the product of a mass in kilograms and a distance in meters, and is commonly used in the context of rotational motion and the study of center of mass.
Law of Conservation of Momentum: The Law of Conservation of Momentum states that the total linear momentum of a closed system remains constant if no external forces are acting on it. This principle is fundamental in analyzing collisions and interactions in mechanics.
Linear Momentum: Linear momentum is a vector quantity that describes the motion of an object. It is defined as the product of an object's mass and its velocity, and it represents the object's quantity of motion. Linear momentum is a fundamental concept in classical mechanics that is crucial for understanding the behavior of moving objects and the conservation of energy.
M\vec{a}_{cm}: M\vec{a}_{cm} is the net force acting on the center of mass of a system, which determines the acceleration of the center of mass. It is a fundamental concept in the study of rigid body dynamics and the motion of systems of particles.
Mass Distribution: Mass distribution refers to the way in which the mass of an object is spread out or distributed throughout its volume. It is a crucial concept in the study of the center of mass, as the distribution of mass within an object affects the location of its center of mass.
Moment of inertia: Moment of inertia is a measure of an object's resistance to changes in its rotational motion about a fixed axis. It depends on the mass distribution relative to the axis of rotation.
Moment of Inertia: The moment of inertia is a measure of an object's resistance to rotational acceleration. It is a scalar quantity that depends on the mass and distribution of an object's mass about a given axis of rotation. The moment of inertia is a crucial concept in the study of rotational dynamics, as it determines how an object will respond to applied torques.
Newton's Laws of Motion: Newton's Laws of Motion are a set of three fundamental principles that describe the relationship between an object and the forces acting upon it, governing the motion of objects in the physical world. These laws form the foundation of classical mechanics and are essential in understanding the behavior of objects in various contexts, including the Scope and Scale of Physics, Algebra of Vectors, Free Fall, Newton's First Law, Impulse and Collisions, and Center of Mass.
Particle System: A particle system is a computer graphics technique used to simulate certain types of dynamic systems, such as fire, smoke, water, explosions, and other natural phenomena. It involves the generation and manipulation of large numbers of individual particles to create complex and realistic visual effects.
Rigid Body: A rigid body is an idealized object that maintains its shape and size regardless of the forces acting upon it. It is a fundamental concept in classical mechanics that simplifies the analysis of the motion and behavior of objects.
Symmetry: Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis. It is a fundamental concept in physics that describes the invariance of a system under certain transformations, such as translations, rotations, or reflections.
Torque: Torque is a measure of the rotational force applied to an object, which causes it to rotate about an axis. It is influenced by the magnitude of the force applied, the distance from the axis of rotation, and the angle at which the force is applied, making it crucial for understanding rotational motion and equilibrium.
X_{cm}: $x_{cm}$ is the x-coordinate of the center of mass of an object or system. The center of mass is the point at which the entire mass of an object can be considered to be concentrated, and it is an important concept in the study of mechanics and dynamics.
Y_{cm}: y_{cm} represents the y-coordinate of the center of mass of a system or object. The center of mass is the point at which the entire mass of the system can be considered to be concentrated, and it is the point around which the system rotates in the absence of external forces.
Z_{cm}: The z-coordinate of the center of mass of an object or system. The center of mass is the point where the entire mass of an object or system can be considered to be concentrated, and it is a crucial concept in classical mechanics for understanding the motion of objects under the influence of forces.
ρ(x, y, z): ρ(x, y, z) represents the mass density function, which describes the distribution of mass within a three-dimensional object or system. It is a fundamental concept in the study of center of mass, as the mass density function is used to calculate the location of the center of mass.
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