What to include when solving a problem

• Picture w/ reference frame

• List given variables

• List what you need to find

• Pure equations (no substitutions)

• Units and unit cancelations

Variables

Radians = $rad$

Chapter 7: Impulse and Momentum

Impulse: the change of momentum of an object when the object is acted upon by a force for an interval of time.

• $\overrightarrow{J}=\overline{\overrightarrow{F}}\Delta t$m ($N\cdot s$)

• Same direction as the average force (vector)

Linear momentum: the product of a system's mass multiplied by its velocity (mass in motion).

• $\overrightarrow{p}=m\overrightarrow{v}$

• SI unit: $\dfrac{kg\cdot m}{s}$

• Has same direction as the velocity

Impulse-momentum theorem: the impulse applied to an object will be equal to the change in its momentum.

• $\left( \sum \overline{\overrightarrow{F}}\right) \Delta t=m\overrightarrow{v}_{f}-m\overrightarrow{v}_{0}$

• Impulse x elapsed time = final momentum - initial momentum

• Apply force to object = object gained momentum.

• Object that loses momentum must be transmitting a force.

• If an object is undergoing a change in momentum, the object is experiencing an impulse.

• No impulse acting on an object (zero) means momentum is conserved.

Principle of conservation of linear momentum: The total linear momentum of an isolated systems remains constant (is conserved) if no external forces act on it.

• $\overrightarrow{p}_{f}=\overrightarrow{p}_{0}$

Isolated system: a system where the sum of external forces is zero.

Internal forces: Forces that objects within system exert of each other (cancel due to Newton’s 3rd law, so they can be ignored).

External forces: Forces exerted on objects by agents outside the system. If 0, momentum is conserved.

Applying the Principle of Conservation of Linear Momentum

1. Decide which objects are included in the system.

2. Identify internal and external forces.

3. Verify system is isolated (no net external forces, forces must sum to zero).

4. Set final momentum of system equal to its initial momentum.

5. Check signs.

In any collisions linear momentum is conserved. Three types of collisions:

Basketball examples.

Collisions in one dimension: $m_{1}v_{f1}+m_{2}v_{f2}=m_{1}v_{01}+m_{2}v_{02}$

Final velocity after collision: $v_{f}=\dfrac{m_{1}v_{01}+m_{2}v_{02}}{m_{1}+m_{2}}$

Collisions (in two dimensions) within isolated systems conserve perpendicular components of momentum.

• x-components: $m_{1}v_{ f1x}+m_{2}v_{f2x}=m_{1}v_{01x}+m_{2}v_{02x}$

• y-components: $m_{1}v_{ f1y}+m_{2}v_{f2y}=m_{1}v_{01y}+m_{2}v_{02y}$

Center of mass: a point that represents the average location for the total mass of a system.

• $x_{cm}=\dfrac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}$

Velocity of center of mass: $v_{cm}=\dfrac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}$

• Total linear momentum of isolated system does not change, so velocity of the center of mass does not change.

Chapter 8: Rotational Kinematics

Angular displacement: the angle at which a body rotates around a center or axis of rotation. Intersects axis of rotation perpendicularly.

• Equation: $\Delta \theta =\theta -\theta _{0}$(radian)

• Counterclockwise is positive.

• Clockwise is negative.

• Angular displacement in radians = Arc length/ Radius

  • Equation: $\theta =\dfrac{s}{r}$ (radians)

• Arc length equation: $s=r\theta$ (meters)

• One full revolution: $\theta =2\pi rad$

• Conversion: 1 rev = $2\pi rad=360^{\circ }$

Average angular velocity = Angular displacement/Elapsed time

• Equation: $\overline{\omega }=\dfrac{\Delta \theta }{\Delta t}$ ($rad/s$)

• Vector quantity

• Counterclockwise rotation is positive

• Clockwise rotation is negative

Average angular acceleration = Change in angular velocity/Elapsed time

• Equation: $\overline{\alpha }=\dfrac{\Delta \omega }{\Delta t}$ ($rad/s^{2}$)

• Vector quantity

The Equations of Kinematics for Rotational Motion

$\alpha$ = constant

$\omega =\omega _{0}+\alpha t$

$\theta =\dfrac{1}{2}\left( \omega _{0}+\omega \right) t$

$\theta =\omega _{0}t+\dfrac{1}{2}\alpha t^{2}$

$\omega ^{2}=\omega _{0}^{2}+2\alpha \theta$

Tangential velocity: the linear speed of an object moving along a circular path at any given point. Radius x angular velocity.

• Equation: $v_{T}=r\omega$($m/s$)

• $\omega$ must be in $rad/s$

• Vector quantity

• When there is no direction, it turns into a tangential speed

• Center of pivot has slowest tangential speed, out has fastest.

Tangential acceleration: how fast the tangential speed is changing.

• Equation: $a_{T}=r\alpha$($m/s^{2}$)

• $\alpha$ must be in $rad/s^{2}$

Centripetal acceleration: the rate of change of an object's tangential velocity when moving in a circular path.

• Equation: $a_{c}=r\omega ^{2}$ ($m/s^{2}$)

Tangential acceleration: how fast the tangential speed is changing.

• Equation: $a_{T}=r\alpha$ ($m/s^{2}$)

• $\alpha$ must be in $rad/s^{2}$

Breaking down into x and y components: pythagorean theorem

• $\overrightarrow{a}=\sqrt{\overrightarrow{a}_{c}^{2}+\overrightarrow{a_{T}}^{2}}$

Rolling motion:

• The tangential speed of a point on the outer edge of the tire is equal to the speed of the car over the ground.

• Angular speed $v=rw$

• Angular acceleration $a=r\alpha$

Chapter 9: Rotational Kinematics

• Pure translational motion: all points on an object move in parallel paths.

• General motion: combination of translation and rotation.

Torque: a turning force causing angular acceleration. Mag of force x lever arm

• Equation: $\tau =Fl$ ($N\cdot m$)

• Torque amount depends on:

  • Where and in what direction the force is applied.

  • Axis of rotation.

• Maximum turning occurs farthest from the axis of rotation.

• Counterclockwise rotation = positive torque

• Lever arm: the distance from the axis of rotation to the line of action of the force.

  • Bigger the lever arm, bigger the torque.

• Line of action: runs through force vector that is creating the torque. Extends in both directions

• Rigid body equilibrium:

  • Zero translational acceleration.

  • Zero angular acceleration.

• Equilibrium conditions:

  • Sum of externally applied forces is zero.

    • $\Sigma F_{x}=0$

    • $\Sigma F_{y}=0$

  • Sum of externally applied torques is zero.

    • $\Sigma \tau =0$

• Reasoning Strategy

  • Choose the object for equilibrium analysis.

  • Draw a free body diagram showing all external forces.

  • Select suitable x, y axes and break forces into components along these axes.

  • Apply equilibrium equations: Set net forces in x and y directions to zero.

  • Choose a convenient axis of rotation. Set sum of torques about this axis to zero.

  • Solve equations for desired unknowns.

Static Equilibrium

• Stillness and Balance: Static equilibrium means something isn't moving and is balanced.

• Force and Torque Balance: To stay still, everything pushing or pulling on the object must balance out. This includes both pushing or pulling forces and torques (like twisting forces).

• Three Conditions: For static equilibrium, three things must be true:

  • All the forces on the object add up to zero.

  • All the twisting forces on the object add up to zero.

  • The object doesn't start moving or rotating on its own.

• Real-Life Examples: Designing stable buildings, bridges, and other structures.

• Stability: Objects can be stably balanced (like a bowl sitting still), unstably balanced (like a pencil standing on its tip), or neutrally balanced (like a ball sitting in a bowl).

Center of gravity: point where an object's weight effectively acts when calculating torque.

• If an object is symmetrical shape and uniform, the center of gravity lies at its geometrical center.

• Equation: $x_{cg}=\dfrac{W_{1}x_{1}+W_{2}x_{2}}{W_{1}+W_{2}}$

  • More can be added on ff more weights are presents.

Tangential force: $F_{T}=ma_{T}$

• With substitution for $F_{T}$ and $a_{T}$: $\tau =\left( mr^{2}\right) \alpha$

Inertia: the tendency of an object to resist changes in its motion.

• Moment of inertia for a single point mass:

  • Equation: $I=mr^{2}$

  • Inertia can only be zero or positive

• Greater moment of inertia corresponds to increased resistance for an object's motion.

• If you have a weird shape being spun around an axis, the moment of inertia of that axis is going to be basically the sum of the moments of inertia of all the point masses that make up that object.

  • Equation: $\Sigma \tau=\sum \left( mr^{2}\right) \alpha$

    • $\tau _{1}{\alpha }$+$\tau _{2}{\alpha }$+$\tau _{3}{\alpha }$

Rotational Analog

Net external torque = (Moment of inertia)x(Angular acceleration)

• Equation: $\Sigma \tau =I\alpha$ ($kg\cdot m^{2}$)

• Requirement: Angular acceleration must be expressed in radians/s²

Rotational work: the energy transformation that occurs when rotating objects experience torque over a distance (results in change in rotational KE). Scalar.

• Equation: $W_{R}=\tau \theta$ ($J$)

  • $\theta =\dfrac{s}{r}$

• Requirement: The angle must be expressed in radians

Rotational kinetic energy: the energy a rigid object has due to its rotation. Scalar.

• Equation: $KE_{R}=\dfrac{1}{2}I\omega ^{2}$ ($J$)

Total kinetic energy = $KE_{tota1}=\dfrac{1}{2}I\omega ^{2}+\dfrac{1}{2}mv_{T}^{2}$

Mechanical energy of a rotating object: $E=\dfrac{1}{2}mv^{2}+\dfrac{1}{2}I\omega ^{2}+mgh$

• Transitional KE + Rotational KE + PE

Energy Conservation: $\dfrac{1}{2}mv_{f}^{2}$ + $\dfrac{1}{2}I\omega _{f}^{2}$ + $mgh_{f}$ = $\dfrac{1}{2}mv_{0}^{2}$ + $\dfrac{1}{2}I\omega _{0}^{2}$ + $mgh_{0}$

Angular momentum: the product of an object's moment of inertia and its angular velocity around the same axis.

• Equation: $L=I\omega$ ($kg\cdot m^{2}/s$)

• Requirement: The angular speed must be expressed in rad/s

Principle of Conservation of Angular Momentum

• The angular momentum of a spinning system remains constant (is conserved) if the net external torque acting on the system is zero.

• Example: Ice skater spinning