Unit 9: Center of Mass and Linear Momentum

9.1 - Center of Mass

Important idea: the center of mass of a system of $n$ particles, with a total mass $M$ is the point with coordinates:

x_{com} = \frac{1}{M} \sum_{i=1}^n m_i x_i

y_{com} = \frac{1}{M} \sum_{i=1}^n m_i y_i

z_{com} = \frac{1}{M} \sum_{i=1}^n m_i z_i

Which can also be represented in vector form:

\vec r_{com} = \frac{1}{M} \sum_{i=1}^n m_i \vec r_i

For a solid mass, such as a ball, which contains a massive number of particles, the object can be treated as a continuous distribution of matter. The "particles" then become differential mass elements $dm$ , and the coordinates for the center of mass can be represented as:

x_{com} = \frac{1}{M} \int x \space dm

y_{com} = \frac{1}{M} \int y \space dm

z_{com} = \frac{1}{M} \int z \space dm

where $M$ is now the mass of the object.

In order to evaluate these integrals with common densities, we use:

\rho = \frac{dm}{dV} = \frac{M}{V}

and substitute into the above equations to finally get:

x_{com} = \frac{1}{V} \int x \space dV

y_{com} = \frac{1}{V} \int y \space dV

z_{com} = \frac{1}{V} \int z \space dV

where $V$ is the total volume of the object.

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Symmetry as a Shortcut: You can bypass one or more of these integrals if an object has a point, a line, or a plane of symmetry. The center of mass of such an object then lies at that point, on that line, or in that plane.

9.2 - Newton's Second Law for a System of Particles

The motion of the center of mass of any system of particles is governed by Newton’s second law for a system of particles, which is:

\vec F_{net} = M\vec a_{com}

where $\vec F_{net}$ is the net force of all external forces acting on the system, $M$ is the total mass of the system, and $\vec a_{com}$ is the acceleration of the system's center of mass.

9.3 - Linear Momentum

For a single particle, we can define a quantity $\vec p$ called its linear momentum as:

\vec p = m \vec v

which is a vector quantity in the same direction as the particle's velocity.

Newton's second law can also be written in terms of momentum:

\vec F_{net} = \frac{d \vec p}{dt}

Systems of Particles

For a system of particles, these relations can be written as the following:

\vec P = M \vec v_{com}

\vec F_{net} = \frac{d \vec P}{dt}

9.4 - Collision and Impulse

Impulse-Linear Momentum Theorem

The change in a body's linear momentum over time during a collision is the impulse, $\vec J$ .

\vec p_f - \vec p_i = \Delta \vec p = \vec J

The impulse can also be written as the integral of the force $\vec F(t)$ exerted on the body in the collision:

\vec J = \int_{t_i}^{t_f} \vec F(t) \space dt

One-Dimensional Motion

If $F_{avg}$ is the average magnitude of $\vec F(t)$ during the collision and $\Delta t$ is the duration of the collision, then for one-dimensional motion the impulse can be simplified as

J = F_{avg} \Delta t

Steady Stream of Collisions with a Fixed Body

When a steady stream of bodies, each with mass $m$ and speed $v$ , collides with a body whose position is fixed, the average force on the fixed body is:

F_{avg} = -\frac{n}{\Delta t} \Delta p = -\frac{n}{\Delta t} m \Delta v

where $n/\Delta t$ is the rate at which the bodies collide with the fixed body, and $\Delta v$ is the change in velocity of each colliding body. The average force can also be written as:

F_{avg} = -\frac{\Delta m}{\Delta t} \Delta v

where $\Delta m / \Delta t$ is the rate at which mass collides with the fixed body. The change in velocity is $\Delta v = -v$ if the bodies stop upon impact and $\Delta v = -2v$ if they bounce backward with no change in their speed.

9.5 - Conservation of Linear Momentum

If a system is closed and isolated so that no net external force acts on it, then the linear momentum P must be constant even if there are internal changes:

\vec P_i = \vec P_f

9.6 - Momentum and Kinetic Energy in Collisions

In an inelastic collision of two bodies, the kinetic energy of the two-body system is not conserved. If the system is closed and isolated, the total linear momentum of the system must be conserved, which we can write in vector form as:

\vec p_{1i} + \vec p_{2i} = \vec P_{2i} + \vec p_{2f}

where the $i$ subscripts are before the collision and the $f$ subscripts are the values right after the collision.

If the motion of the bodies is along a single axis, the collision is one-dimensional and we can write the equation in terms of velocity components along that axis:

m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}

Bodies Sticking Together

If the bodies colliding stick together after the collision, then the collision is completely inelastic and the bodies have the same final velocity $V$ .

Center of Mass

The center of mass of a closed, isolated system of two colliding bodies is not affected by a collision. In particular, the velocity $\vec v_{com}$ of the center of mass cannot be changed by the collision.

9.7 - Elastic Collisions in One Dimension

An elastic collision is a special type of collision in which the kinetic energy of a system of colliding bodies is conserved. If the system is closed and isolated, its linear momentum is also conserved. For a one-dimensional collision in which body 2 is a target and body 1 is an incoming projectile, conservation of kinetic energy and linear momentum yield the following expressions for the velocities immediately after the collision:

v_{1f} = \frac{m_1-m_2}{m_1+m_2} v_{1i}

v_{2f} = \frac{2m_1}{m_1+m_2} v_{1i}

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You don't necessarily have to memorize these, they are simply the result of the conservation of the kinetic energy of the system during the collision.

9.8 - Collisions in Two Dimensions

If two bodies collide and their motion is not along single axis (the collision is not straight on), the collision is two-dimensional. If the two-body system is closed and isolated, the law of conservation of momentum applies and can be written as:

\vec P_{1i} + \vec P_{2i} = \vec P_{1f} + \vec P_{2f}

In the case that the collision is also elastic, then the total kinetic energy is also conserved:

K_{1i} + K_{2i} = K_{1f} + K_{2f}

9.9 - Systems with Varying Mass: A Rocket