Kinetics of a Particle: Impulse and Momentum Study Notes

Principle of Linear Impulse and Momentum

• Development of the Principle: The principle is derived by integrating the equation of motion for a particle with respect to time. Starting with Newton’s Second Law: $\sum \mathbf{F} = m\mathbf{a} = m \frac{d\mathbf{v}}{dt}$. Rearranging the terms yields $\sum \mathbf{F} \, dt = m \, d\mathbf{v}$. Integrating between the limits $\mathbf{v} = \mathbf{v}_1$ at $t = t_1$ and $\mathbf{v} = \mathbf{v}_2$ at $t = t_2$ results in: $m\mathbf{v}_1 + \sum \int_{t_1}^{t_2} \mathbf{F} \, dt = m\mathbf{v}_2$.

• Linear Momentum ($\mathbf{L}$): Defined as the vector quantity $\mathbf{L} = m\mathbf{v}$.     * Since mass ($m$) is a positive scalar, the momentum vector $\mathbf{L}$ maintains the same direction as the velocity vector ($\mathbf{v}$).     * Units: Expressed in units of mass-velocity, such as $\text{kg} \cdot \text{m/s}$ or $\text{slug} \cdot \text{ft/s}$.

• Linear Impulse ($\mathbf{I}$): Defined as the integral $\mathbf{I} = \int \mathbf{F} \, dt$.     * This vector measures the cumulative effect of a force over the duration it acts.     * It acts in the direction of the force.     * Units: Expressed in units of force-time, such as $\text{N} \cdot \text{s}$ or $\text{lb} \cdot \text{s}$.     * Constant Force: For a force constant in both magnitude and direction ($\mathbf{F}_c$), the impulse simplifes to: $\mathbf{I} = \mathbf{F}_c(t_2 - t_1)$.     * Variable Force: Graphically, the magnitude of the impulse is represented as the area under the force-versus-time curve.

• Application in Dynamics: The principle $\text{Initial Momentum} + \text{Sum of Impulses} = \text{Final Momentum}$ provides a direct method for finding final velocities when forces and time intervals are known. This is a one-step alternative to the two-step process of finding acceleration via $\sum \mathbf{F} = m\mathbf{a}$ and then integrating $a = d\mathbf{v}/dt$.

• Scalar Equations: Resolve the vector equation into three scalar components:     * $m(v_x)_1 + \sum \int_{t_1}^{t_2} F_x \, dt = m(v_x)_2$     * $m(v_y)_1 + \sum \int_{t_1}^{t_2} F_y \, dt = m(v_y)_2$     * $m(v_z)_1 + \sum \int_{t_1}^{t_2} F_z \, dt = m(v_z)_2$

Linear Impulse and Momentum for a System of Particles

• Equation of Motion for Systems: For a collection of particles, the sum of external forces is related to the derivative of the total momentum: $\sum \mathbf{F}_i = \sum m_i \frac{d\mathbf{v}_i}{dt}$. Internal forces ($\mathbf{f}_i$) cancel out due to Newton’s Third Law (equal and opposite collinear pairs).

• Integrative Principle: The initial total linear momenta plus the sum of external impulses equals the final total linear momenta: $\sum m_i(\mathbf{v}_i)_1 + \sum \int_{t_1}^{t_2} \mathbf{F}_i \, dt = \sum m_i(\mathbf{v}_i)_2$.

• Mass Center ($G$): The total linear momentum is equivalent to the momentum of a ‘fictitious’ aggregate particle of total mass $m = \sum m_i$ moving at the velocity of the center of mass ($\mathbf{v}_G$). The equation becomes: $m(\mathbf{v}_G)_1 + \sum \int_{t_1}^{t_2} \mathbf{F}_i \, dt = m(\mathbf{v}_G)_2$.

Conservation of Linear Momentum for Systems

• Definition: Linear momentum is conserved when the sum of external impulses acting on a system is zero. Mathematically: $\sum m_i(\mathbf{v}_i)_1 = \sum m_i(\mathbf{v}_i)_2$.

• Condition of Zero External Impulse: Occurs when no external forces act or when external forces are nonimpulsive.

• Impulsive vs. Nonimpulsive Forces:     * Impulsive Forces: Large forces acting over a very short time interval ($\Delta t$) that cause significant momentum changes (e.g., explosions or collisions).     * Nonimpulsive Forces: Small forces relative to impulsive ones, such as weight, air resistance, or low-stiffness springs, whose effect during a very short $\Delta t$ is negligible.

• Center of Mass Velocity: If linear momentum is conserved, the velocity of the mass center ($\mathbf{v}_G$) remains constant throughout the motion study.

Mechanics of Impact: Central and Oblique

• Impact Definition: A collision between two bodies over a very short duration, generating large impulsive forces.

• Types of Impact:     * Central Impact: The directions of motion for both mass centers lie along the line of impact (the line perpendicular to the plane of contact).     * Oblique Impact: The motion of one or both particles occurs at an angle to the line of impact.

• The Coefficient of Restitution ($e$): An experimental value relating the relative separation velocity to the relative approach velocity: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$.     * Elastic Impact ($e = 1$): No energy loss; deformation impulse equals restitution impulse.     * Plastic/Inelastic Impact ($e = 0$): Maximum energy loss; bodies stick together and move at a common velocity.

• Oblique Impact Analysis Procedure:     1. Establish the $x$-axis along the line of impact and the $y$-axis along the plane of contact.     2. Conservation of momentum for the system in the $x$ direction: $\sum m(v_x)_1 = \sum m(v_x)_2$.     3. Coefficient of restitution applies along the line of impact ($x$ direction): $e = \frac{(v_{Bx})_2 - (v_{Ax})_2}{(v_{Ax})_1 - (v_{Bx})_1}$.     4. Conservation of momentum for each individual particle in the $y$ direction (since no impulse acts perpendicular to the line of impact): $(v_{Ay})_1 = (v_{Ay})_2$ and $(v_{By})_1 = (v_{By})_2$.

Angular Momentum and Relation to Moment of a Force

• Scalar Definition: The angular momentum ($H_O$) of a particle about point $O$ is the moment of its linear momentum: $(H_O)_z = d \cdot (mv)$.     * Here, $d$ is the perpendicular distance (moment arm) from $O$ to the line of action of $mv$.

• Vector Definition: Defined via cross product: $\mathbf{H}_O = \mathbf{r} \times m\mathbf{v}$.     * Can be evaluated via determinant: $\mathbf{H}_O = \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ r_x & r_y & r_z \\ m v_x & m v_y & m v_z \end{pmatrix}$.

• Relation to Moments: The resultant moment about point $O$ ($\sum \mathbf{M}_O$) equals the time rate of change of the angular momentum ($\dot{\mathbf{H}}_O$): $\sum \mathbf{M}_O = \dot{\mathbf{H}}_O$.     * For a system of particles: $\sum \mathbf{M}_O = \sum (\mathbf{r}_i \times \mathbf{F}_i) = \frac{d}{dt} \sum (\mathbf{r}_i \times m_i \mathbf{v}_i)$.

Principle of Angular Impulse and Momentum

• The Principle: Integrating the moment equation over time: $(\mathbf{H}_O)_1 + \sum \int_{t_1}^{t_2} \mathbf{M}_O \, dt = (\mathbf{H}_O)_2$.     * Angular Impulse: The integral $\int \mathbf{M}_O \, dt = \int (\mathbf{r} \times \mathbf{F}) \, dt$.

• Conservation of Angular Momentum: If the sum of angular impulses about an axis is zero, the angular momentum about that axis is constant: $(\mathbf{H}_O)_1 = (\mathbf{H}_O)_2$.     * Common in central force problems where the force always passes through the center of rotation (moment arm is zero).

Steady Flow of a Fluid Stream

• Control Volume Logic: Analyzing fluid particles entering and exiting a region. Flow is steady if the rate of mass entering equals the rate exiting.

• Force Equilibrium: The resultant force $\sum \mathbf{F}$ on a control volume equals the mass flow rate multiplied by the change in velocity: $\sum \mathbf{F} = \frac{dm}{dt}(\mathbf{v}_{out} - \mathbf{v}_{in})$.

• Moment Equilibrium: $\sum \mathbf{M}_O = \frac{dm}{dt}(\mathbf{r}_{out} \times \mathbf{v}_{out} - \mathbf{r}_{in} \times \mathbf{v}_{in})$.

• Mass Flow Rate ($\frac{dm}{dt}$): Related to fluid density ($\rho$), velocity ($v$), and cross-sectional area ($A$): $\frac{dm}{dt} = \rho v A$.

• Discharge/Volumetric Flow ($Q$): $Q = v A$. Measured in $dm^3/s$ or $ft^3/s$.

Propulsion and Variable Mass Mechanics

• Control Volumes with Changing Mass: Used for rockets (losing mass) or scoops (gaining mass).

• Losing Mass (Rocket Propulsion): The equation of motion is: $\sum \mathbf{F}_{cv} = m \frac{d\mathbf{v}}{dt} - \mathbf{v}_{D/e} \frac{dm_e}{dt}$.     * $\sum \mathbf{F}_{cv}$: External forces (drag, weight).     * $m \frac{d\mathbf{v}}{dt}$: Mass of the device times its acceleration ($m \cdot a$).     * $\mathbf{v}_{D/e} \frac{dm_e}{dt}$: The thrust ($T$), where $\mathbf{v}_{D/e}$ is the velocity of the ejected mass relative to the device.

• Gaining Mass: The equation of motion is: $\sum \mathbf{F}_{cv} = m \frac{d\mathbf{v}}{dt} + \mathbf{v}_{D/i} \frac{dm_i}{dt}$.     * $\mathbf{v}_{D/i}$: Velocity of the injected mass relative to the device.     * The term $\mathbf{v}_{D/i} \frac{dm_i}{dt}$ represents the retarding force ($R$) resisting motion as mass is gathered.

Questions & Discussion

• Example 15.1: 100-kg stone with constant force:     * Initial state: at rest. $\mathbf{F} = 200 \, \text{N}$ at $45^{\circ}$, time $t = 10 \, \text{s}$, $\mu_k = 0$.     * Using $\sum m(v_{x})_1 + \sum \int F_x \, dt = m(v_{x})_2$:         $0 + 200 \cos(45^{\circ}) \cdot 10 = 100 \cdot v_2 \implies v_2 = 14.1 \, \text{m/s}$.     * Normal force calculation ($\sum F_y = 0$): $N_c(10) - 981(10) + 200 \sin(45^{\circ}) \cdot 10 = 0 \implies N_c = 840 \, \text{N}$.

• Example 15.4: Boxcar and Tank Car Collision:     * Boxcar A ($15 \, \text{Mg}$, $1.5 \, \text{m/s}$, $+x$) and Tank car B ($12 \, \text{Mg}$, $0.75 \, \text{m/s}$, $-x$).     * Coupling means $v_2$ is shared.     * Conservation of momentum: $(15000)(1.5) - (12000)(0.75) = (27000)v_2 \implies v_2 = 0.5 \, \text{m/s}$.     * Average Force ($F_{avg}$) over $0.8 \, \text{s}$: $(15000)(1.5) - F_{avg}(0.8) = (15000)(0.5) \implies F_{avg} = 18.8 \, \text{kN}$.

• Example 15.14: Ball on a Cord with constant radial speed:     * Initially: $r_1 = 1.75 \, \text{ft}$, $v_1 = 4 \, \text{ft/s}$. Cord pulled at $v_c = 6 \, \text{ft/s}$.     * Final: $r_2 = 0.6 \, \text{ft}$.     * Conservation of angular momentum: $r_1 m v_1 = r_2 m v_{\theta} \implies (1.75)v_1 = (0.6)v_{\theta}$.     * $(1.75)(4) = (0.6)v_{\theta} \implies v_{\theta} = 11.67 \, \text{ft/s}$.     * Total final speed: $v_2 = \sqrt{11.67^2 + 6^2} = 13.1 \, \text{ft/s}$.