December 2017 - Page 6 of 8 - Shawn Zhong - 钟万祥

7.1 – Rotational Kinematics

Radians and Degrees • In degrees, once around a circle is 360° • In radians, once around a circle is 2π • A radian measures a distance around an arc equal to the length of the arc's radius • Δs=C=2πr Linear vs. Angular Displacement • Linear position / displacement given by Δr or Δs • Angular position / displacement given by Δθ • s=rθ • Δs=rΔθ Linear vs. Angular Velocity • Linear speed / velocity given by v ⃗ • Angular speed / velocity given by ω ⃗ • v ⃗=(ds ⃗)/dt • ω ⃗=(dθ ⃗)/dt Direction of Angular Velocity Converting Linear to Angular Velocity Linear vs. Angular Acceleration • Linear acceleration is given by a ⃗ • Angular acceleration is given by α ⃗ • a ⃗=(dv ⃗)/dt • α ⃗=(dω ⃗)/dt Kinematic Variable Parallels Variable Translational Angular Displacement Δs Δθ Velocity v ⍵ Acceleration a ⍺ Time t t Variable Translations Variable Translational Angular Displacement s=rθ θ=s/r Velocity v=rω ω=v/r Acceleration a=rα α=a/r Time t=t t=t Kinematic Equation Parallels Centripetal Acceleration Example: Wheel in Motion • A wheel of radius r and mass M undergoes a constant angular acceleration of magnitude ⍺. • What is the speed of the wheel after it has completed on complete turn, assuming it started from rest? 2003 Free Response Question 3 2014 Free Response Question 2

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7.2 – Moment of Inertia

Types of Inertia • Inertial mass / translational inertia (M) is an object's ability to resist a linear acceleration • Moment of Inertia / rotational inertia (I) is an object's resistance to a rational acceleration • Objects that have most of their mass near their axis of rotation have smaller rotational inertias than objects with more mass farther from their axis of rotation Kinetic Energy of a Rotating Disc Calculating Moment of Inertia (I) • I=∑▒〖mr^2 〗=∫▒〖r^2 dm〗 Example 1: Point Masses • Find the moment of inertia (I) of two 5-kg bowling balls joined by a meter-long rod of negligible mass when rotated about the center of the rod. • Compare this to the moment of inertia of the object when rotated about one of the mass Example 2: Uniform Rod • Find the moment of inertia of a uniform rod about its end and about its center Example 3: Solid Cylinder • Find the moment of inertia of a uniform solid cylinder about its axis Parallel Axis Theorem (PAT) • If the moment of inertia (I) of any object through an axis intersecting the center of mass of the object is l, you can find the moment of inertia around any axis parallel to the current axis of rotation (l') Example 4: Calculating I Using PAT • Find the moment of inertia of a uniform rod about its end Example 5: Hollow Sphere • Calculate the moment of inertia of a hollow sphere with a mass of 10 kg and a radius of 0.2 meter Example 6: Adjusting Moment of Inertia • A uniform rod of length L has moment of inertia I0 when rotated about its midpoint. • A sphere of mass M is added to each end of the rod. • What is the new moment of inertia of the rod/ball system?

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7.3 – Torque

Torque (τ) • Torque is a force that causes an object to turn • Torque must be perpendicular to the displacement to cause a rotation • The further away the force is applied from the point of rotation, the more leverage you obtain, so this distance is known as the lever arm (r) • τ ⃗=r ⃗×F ⃗ • |τ ⃗ |=rF sin⁡θ Direction of the Torque Vector • The direction of the torque vector is perpendicular to both the position vector and the force vector • You can find the direction using the right-hand rule. Point the fingers of your right hand in the direction of the line of action, and bend you fingers in the direction of the force • You thumb then points in the direction of your torque • Note that positive torques cause counter-clockwise rotation, and negative torques cause clockwise rotation Newton's Second Law: Translational vs. Rotational • F ⃗_net=ma ⃗ • τ ⃗=Iα ⃗ Equilibrium • Static Equilibrium implies that the net force and the net torque are zero, and the system is at rest • Dynamic Equilibrium implies that the net force and the net torque are zero, and the system is moving at constant translational and rotational velocity Example 1: See-Saw Problem • A 10-kg tortoise sits on a see-saw 1 meter from the fulcrum. • Where must a 2-kg hare sit in order to maintain static equilibrium? • What is the force on the fulcrum? Example 2: Beam Problem • A beam of mass M and Length L has a moment of inertia about its center of 1/12 〖ML〗^2. The beam is attached to a frictionless hinge at an angle of 45° and allowed to swing freely. • Find the beam's angular acceleration Example 3: Pulley with Mass • A light string attached to a mass m is wrapped around a pulley of mass mp and radius R. Find the acceleration of the mass Example 4: Net Torque • A system of three wheels fixed to each other is free to rotate about an axis through its center. Forces are exerted on the wheels as shown. What is the magnitude of the net torque on the wheels? Example 5: Café Sign • A 3-kg café sign is hung from a 1-kg horizontal pole as shown. A wire is attached to prevent the sign from rotating. • Find the tension in the wire 2008 Free Response Question 2

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7.4 – Rotational Dynamics

Conservational of Energy Example 1: Disc Rolling Down an Incline • Find the speed of a disc of radius R which starts at rest and rolls down an incline of height H Rotational Dynamics Example 2: Strings with Massive Pulleys • Two blocks are connected by a light string over a pulley of mass mp and radius R. • Find the acceleration of mass m2 if m1 stis on a frictionless surface Example 3: Rolling Without Slipping • A disc of radius R rolls down an incline of angle θ without slipping. • Find the force of friction on the disc Example 4: Rolling with Slipping • A bowling ball of mass M and radius R skids horizontally down the alley with an initial velocity of v0. Find the distance the ball skids before rilling given a coefficient of kenetic friction μk Example 5: Amusement Park Swing • An amusement park ride of radius x allows children to sit in a spinning swing held by a cable of length L. • At maximum angular speed, the cable makes an angle of θ with the vertical as shown in the diagram below • Determine the maximum angular speed of the rider in terms of g, θ, x and L. 2002 Free Response Question 2 2006 Free Response Question 3 2010 Free Response Question 2 2013 Free Response Question 3

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7.5 – Angular Momentum

Linear Momentum • Momentum is a vector describing how difficult it is to stop a moving object • Total momentum is the sum of individual momenta • p ⃗=mv ⃗ • Units are kg·m/s or N·s Angular Momentum • Angular momentum (L ⃗) is a vector describing how difficult it is to stop a rotating object • Total angular momentum is the sum of individual angular momenta • A mass with velocity v ⃗ moving at some position r ⃗ about point Q has angular momentum L ⃗_Q • Units are kg·m2/s Calculating Angular Momentum • L ⃗_Q=r ⃗×p ⃗=r ⃗×mv ⃗=(r ⃗×v ⃗ )m Spin Angular Momentum • For an object rotating about its center of mass • L ⃗=Iω ⃗ • This is known as an object's spin angular momentum • Spin angular momentum is constant regardless of your reference point Example 1: Object in Circular Orbit • Find the angular momentum of a planet orbiting the sun. Assume a perfectly circular orbit Example 2: Angular Momentum of a Point Particle • Find the angular momentum for a 5-kg point particle located at (2,2) with a velocity of 2 m/s east • About point O at (0,0) • About point P at (2,0) • About point Q at (0,2) Angular Momentum and Net Torque Conservation of Angular Momentum • Spin angular momentum, the product of an object's moment of inertia and its angular velocity about the center of mass, is conserved in a closed system with no external net toques applied • L ⃗=Iω ⃗ Example 3: Ice Skater Problem • An ice skater spins with a specific angular velocity. She brings her arms and legs closer to her body, reducing her moment of inertia to half its original value. What happens to her angular velocity? What happens to her rotational kinetic energy? Example 4: Combining Spinning Discs • A disc with moment of inertia 1 km·m2 spins about an axle through its center of mass with angular velocity 10 rad/s. An identical disc which is not rotating is slide along the axle until it makes contact with the first disc. • If the two discs stick together, what is their combined angular velocity? Example 5: Catching While Rotating • Angelina spins on a rotating pedestal with an angular velocity of 8 radians per second. • Bob throws her an exercise ball, which increases her moment of inertia from 2 kg·m2 to 2.5 kg·m2 • What is Angelina's angular velocity after catching the exercise ball? (Neglect any external torque from the ball) 2005 Free Response Question 3 2014 Free Response Question 3

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