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Translational motion is movement in a straight line Rotational motion is about an axis >Rotation is about an internal axis (earth spins) >Revolution is about an external axis (earth orbits) Radian (θ) measure…ratio of arc length (s) to radius r. When s = r, we have defined 1 radian. r θ s Δs = r Δθ ds d r dt dt dv d r dt dt at ac Total acceleration, Direction for ω and α RHR: direction is along axis of rotation - curl fingers along direction of spin and direction of thumb is direction of ω and α. Explanation is in your book as to why…it stems from mathematical definition of vector in rotation. If object is speeding up, α is in direction of ω If object is slowing down, α is opposite ω Constant acceleration equations: Linear: v = vo +at v2 = vo2 + 2aΔx Δx = vot + ½at2 Rotation for fixed axis: ω = ωo + αt ω2 = ωo2 + 2αΔθ Δθ = ωot + ½αt2 A disc rotates about an axis thru its center according to the equation: 1 3 (t ) t 6t 3 A) Find the angular velocity and acceleration for general time, t. B) Find the mag. of total linear acc. of a point 0.5m from center at t = 1s. C)Find the linear speed of a point on disc 20cm from center at t = 2s Rotational Inertia and Kinetic energy The kinetic energy of a rotating object will be the sum of the kinetic energy of every point on the object: K 12 m1v12 12 m2 v 22 12 m3 v32 ... n n K m v mi r i 1 1 2 2 i i i 1 1 2 2 i n 2 2 1 K 2 mi ri i ( is constant so it can be pulled out) i 1 we define the rotational inertia (or moment of inertia) as where I is measured in kg*m2 n I mi ri i 1 2 Rotational Inertia for a SYSTEM: A uniform rod of length L and mass M that can pivot about its center of mass has 2 masses m1 and m2 placed at each end. The moment of inertia of a uniform rod about its center is (1/12) ML2. Find moment of inertia for the system. L m1 m2 Calculus to find I To sum up the infinite points on a solid object you must integrate the equation for rotational inertia (I) What is the rotational inertia of a rod of length L and linear mass density, λ, spinning around an axis through its center of mass? Parallel Axis Theorem It is used when you already know I of body about an axis that is parallel to another axis you are trying to find. What is the moment of inertia for a uniform rod of length L and mass M spinning halfway between the center of mass & its end? Rotational kinetic energy If object has only translational motion then its kinetic energy is just K trans mv 1 2 2 If an object has only rotational motion then its kinetic energy is just If it has both translational & rotational motion then its kinetic energy is A uniform rod of mass m and length L can rotate about a frictionless hinge that is fixed. If the rod is released from rest where it rotates downward about the hinge, find tangential & angular speed of the edge of the rod at the bottom of the swing. Cross Product The cross product is a vector product (recall dot product was a scalar product). The cross product of two vectors produces a third vector which is perpendicular to the plane in which the first two lie. That is, for the cross of two vectors, A and B, we place A and B so that their tails are at a common point (tail to tail). Their cross product, A x B, gives a third vector, C, whose tail is also at the same point as those of A and B. The vector C points in a direction perpendicular (or normal) to both A and B. The cross product is defined by the formula A x B = |AB|sinθ î × î = ĵ × ĵ = k×k = (1)(1)(sin 0°) = 0 Newton’s 2nd Law for Rotation Torque can cause a change in rotational motion or can cause a rotational acceleration. The distance from the pivot that the force acts is called the leverarm or moment arm, r. r O F Rotational Equilibrium Fy 0 Fx 0 0 Example: A hungry 700N bear walks out on a uniform beam in an attempt to retrieve some goodies hanging at the end. The beam weighs 200N and is 6.0m long; the goodies weigh 80N. a) Draw a force diagram. b) When the bear is at 1.00 m, determine the tension in the wire. c) Determine angle of the reaction force from wall on beam. A rod is held in place by a light wire attached to a wall as shown. The weight of rod is 1000N. Hanging from rod is 2000N crate. a) Find the tension in the wire. Diagram forces. b) Determine reaction force (mag & dir) Example 3: A ladder having a uniform density and a mass, m, rests against a frictionless vertical wall at an angle of 60◦ . The lower end rests on a flat surface where the coefficient of static friction is µs = 0.40. A student with a mass M = 2m attempts to climb the ladder. What fraction of the length L of the ladder will the student have reached when the ladder begins to slip? Draw a force diagram. Two masses hang over a fixed pulley. The pulley has mass 1.5 kg & radius 15cm where m1=15 kg and m2 = 10kg. The rope moves through the pulley without slipping. A. What is the acceleration of the boxes? m1 m2 B. Determine the two tensions. Yo-Yo A string is wound around a Yo-Yo of mass M and radius R. The Yo-Yo is released and allowed to fall from rest. Find acceleration and tension in string as it falls. Make rotation equation and force equation for Yo-Yo. Work & Power Work done on an object can change either its translational kinetic energy or rotational kinetic energy or both ANGULAR MOMENTUM Consider a particle of mass, m, instantaneous velocity, v, and position vector, r, where particle moves in the xy plane about origin O. y v r O m x The particle therefore has momentum, p=mv. We extend the position vector, r, to see the angle between r and p. Angular Momentum of a particle moving about point O, is defined as: y p θ r θ O rsinθ x Direction of L is out of page using RHR. Continuing with the formula for angular momentum… Assuming the angle between r and p is 90o then L = rmv L = rm(rω) (using our linear-angular conversion) L = mr2ω where I=mr2, so we now get L L=Iω NOTE: An object can possess angular momentum about any point, regardless if it’s moving in a circle, orbit, or line about some point. In the figure to left, a dropped object can have angular momentum about the origin where r is increasing along with the velocity and therefore the angular momentum. Torque and Angular Momentum net I Conservation of Angular Momentum Assuming no net external torques Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system Consider the next example in regards to torque and angular momentum vectors EXAMPLE 1 Consider a thin rod of mass, M, and length, L, lying on a frictionless table. There is a frictionless pivot at the top end of the rod A mass, m, slides in a speed, vo, and collides with the rod a distance 2/3L from pivot. The mass rebounds with speed, ¼vo, where moment of inertia of rod is 1/12ML2 about CM. Top View Find the angular velocity after the collision EXAMPLE 2 A dart of mass, m, is shot with speed, vo, at a hoop of mass, M, and radius R. The hoop can be considered a ring. The dart strikes and sticks into top of hoop. a) Find speed that wheel and dart rotate with after collision. b) Find KE lost due to collision. ROLLING When a rigid object rolls across a perfectly level surface, the object’s contact point with the surface is instantaneously at rest. If this were untrue, the object would be slipping or skidding. Because the contact point is at rest, you can think of this as an instantaneous axis of rotation. Relative to the contact point, all points on the object have the same angular velocity even though they have different linear velocities. You can think of rolling as a combination of a pure rotation about the CM and a pure translation of the CM. The velocity of any point on the disk as seen by an observer on the ground is the vector sum of the velocity with respect to the center of mass and the velocity of the center of mass with respect to the ground: v pt on disk rel ground = v pt on disk rel cm + v cm rel to ground FOR EXAMPLE… vtop rel to cm R vCM rel to grnd R vtop rel to grnd 2vCM The point on the top of the wheel has a speed (rel to ground) that is twice the velocity of the center of mass. Consider the point in contact with the ground: vbot rel to cm R vCM rel to grnd R vbot rel to grnd R R 0 The point in contact with the ground has a speed of zero, momentarily at rest. If your car is traveling down the highway at 70 mph, the tops of your wheels are going 140 mph while the bottoms of the wheels are going 0 mph. Rolling Motion is considered a combination of both translational and rotational Consider a solid sphere rolling from rest down an incline. Question: What forces would act on the rolling sphere? Would friction have to act for it to roll? Example: A solid sphere rolls down an incline. Determine aCM and the static frictional force on the sphere of mass, m, & radius R. Rolling without Slipping If object rolls without slipping, the arc-length of a path along the surface of the object as it rotates matches the translational distance traveled by the center of the object. Since s = Rθ Differentiating both sides with respect to time (ds/dt = Rdθ/dt, where R=constant) yields vcm = Rω (R = distance from pt of contact) acm = Rα **These are the conditions for rolling without slipping or smooth rolling. Violating pure rolling condition: If a force were applied to CM of a sphere (no friction), the sphere would start to move, changing vCM, but without changing ω. F Static friction needs to be present to initiate rolling. In pure translation like in sliding, the sole purpose of friction is to oppose relative motion between surfaces. In the case of rolling, friction converts a part of one type of acceleration to another (from linear to angular as in this case). Probably more apropos would be saying that friction changes a part of translational kinetic energy into rotational kinetic energy. Without friction, the force passing through the CM would have only caused linear acceleration. Example 1: A solid cylinder is at rest on a flat surface. When a horizontal force, F, is exerted on the cylinder’s axle, what is the minimum coefficient of static friction to keep the cylinder from slipping? Cylinder has mass, m. F fs Why would friction be to the left? Example 2: A cylinder of mass M and radius R has a string wrapped around it, with the string coming off the cylinder above the cylinder. If the string is pulled to the right with a force F, what is the acceleration of the cylinder if the cylinder rolls without slipping? What is the frictional force acting on the cylinder? F QUESTION: Must there be friction? If so, what kind? What direction? Rolling with Slipping Example 2: A bowling ball of mass, m, and radius, R, is initially thrown so that it only slides with speed vo but doesn’t rotate. As it slides, it begins to spin, and eventually rolls without slipping. How long will it take to stop sliding with only pure rolling without slipping?