Shawn Zhong

Shawn Zhong

 钟万祥
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Shawn Zhong

 钟万祥
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Onetone Blog

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5.3 – Center of Mass

  • Dec 08, 2017
  • Shawn
  • AP Physics C Mechanics
  • No comments yet
Center of Mass • Real objects are more complex than theoretical particles • Treat entire object as if its entire mass were contained at a single point known as the object's center of mass (CM) • Center of mass is the weighted average of the location of mass in an object Find CM by Inspection • For uniform density objects, CM is the geometric center • For objects with multiple parts, find CM of each part and treat as a point • For irregular objects, suspend object from two or more points and drop a plumb line. The line intersects at the center of mass Calculating CM for Systems of Particles • r_CM=(∑▒〖mr ⃗ 〗)/(∑▒m) • x_CM=(m_1 x_1+m_2 x_2+…)/(m_1+m_2+…) • y_CM=(m_1 y_1+m_2 y_2+…)/(m_1+m_2+…) Example 1: Center of Mass (1D) Example 2: CM of Continuous System Example 3: Center of Mass (2D) Finding CM by Integration • For more complex objects, you can find the center of mass by summing up all the little pieces of position vectors multiplied by the differential of mass and dividing by the total mass • r ⃗_CM=(∫▒〖r ⃗dm〗)/M Example 4: CM of a Uniform Rod Example 5: CM of a Non-Uniform Rod • Find the center of mass of a non-uniform rod of length L and mass M whose density is given by λ=kx Center of Mass Relationships Center of Gravity • Center of Gravity refers to the location at which the force of gravity acts upon an object as if it were a point particle with all its mass focused at that point • In a uniform gravitational field, Center of Gravity and Center of Mass are the same • In a non-uniform gravitational field, they may be different 2004 Free Response Question 1
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6.1 – Uniform Circular Motion

  • Dec 08, 2017
  • Shawn
  • AP Physics C Mechanics
  • No comments yet
Uniform Circular Motion • Object travels in a circular path at constant speed • Distance around the circle is its circumference ○ C=2πr=πd • Average speed formula from kinematics still applies ○ v ̅=d/t=2πr Frequency • Frequency is the number of revolutions or cycles which occur each second • Symbol is f • Units are 1/s, or Hertz (Hz) • f=(number of cycles)/second=(number of revolution)/second Period • Period is the time it takes for one complete revolution, or cycle. • Symbol is T • Unites are seconds (s) • T = time for 1 cycle = time for 1 revolution Frequency and Period • f=1/T • T=1/f Centripetal Acceleration • Is an object undergoing UCM accelerating? • Magnitude of Centripetal Acceleration ○ a_c=v^2/r Centripetal Force • If an object is traveling in a circle it is accelerating toward the center of the circle • For an object to accelerate, there must be a net force • We call this force a centripetal force (F_c) • F_c=(mv^2)/r
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7.1 – Rotational Kinematics

  • Dec 08, 2017
  • Shawn
  • AP Physics C Mechanics
  • No comments yet
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

  • Dec 08, 2017
  • Shawn
  • AP Physics C Mechanics
  • No comments yet
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

  • Dec 08, 2017
  • Shawn
  • AP Physics C Mechanics
  • No comments yet
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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