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Today: (Ch. 3) Forces and Motion in Two and Three Dimensions Equilibrium and Examples Tomorrow: (Ch. 4) Projectile Motion Example A passenger weighing 598 N rides in an elevator. The gravitational field strength is 9.8 N/kg. What is the apparent weight of the passenger in each of the following situations? In each case the magnitude of elevator’s acceleration is 0.5 m/s2. (a)The passenger is on the 1st floor and has pushed the button for the 15th floor i. e. the elevator is beginning to move upward. (b)The elevator is slowing down as it nears the 15th floor. Statics and Equilibrium • Statics Deals with objects at rest • Statics is an area of mechanics dealing with problems in which both the velocity and acceleration are zero • The object is also said to be in translational equilibrium Often the “translational” is dropped Equilibrium and Newton’s law According to Newton’s 2nd law F ma This vector quantity can be written in terms of two perpendicular components F x max F y may When object in equilibrium the net force acting on it is zero: For object to be in equilibrium F x max 0 and F y may 0 Equilibrium on an inclined plane N Fa h Identify all the forces in the d system φ W mg y x Choose the axes, which are convenient to work with… Wy φ Wx W φ Using Newton’s Second Law • Newton’s Second Law F m a • Determine all the individual forces acting on the object • Construct a free body diagram • Add the individual forces as vectors • Use Newton’s Second Law to find the acceleration • Acceleration can be used to determine velocity and displacement Equilibrium Example • Forces acting in A – Gravity and normal force in ydirection – Force exerted by person (push) and static friction in x-direction • Free body diagram in B • Forces in x and y components and apply condition of equilibrium ΣFx = 0 and ΣFy = 0 For y-direction: N - m g = 0 For x-direction: Fpush - Ffriction = 0 Equilibrium Example 2 • All the forces do not all align with the x- or y-axes • Find the x- and y-components of all forces that are not on an axis • Applying Newton’s Second Law: – ΣFx = Tx – Ffriction = T cos θ – Ffriction = 0 – ΣFy = N – mg + Ty = N – mg + T sin θ = 0 Equilibrium Example 3 • Both sections of the rope exert a tension force at the center where the walker is standing • The walker and the rope are at rest • The forces acting at the center – Tension on the right and on the left & Weight of the walker Problem Solving Strategy for Statics Problems • Recognize the principle – For static equilibrium, the sum of the forces must be zero – Use • Sketch the problem – Show the given information in the picture – Include a coordinate system • Identify the relationships – Use all the forces to construct a free body diagram – Express all the forces on the object in terms of their x- and y-components – Apply ΣFx = 0 and ΣFy = 0 – May also include ΣFz = 0 Problem Solving Strategy for Statics Problems, cont. • Solve – Solve all the equations – The number of equations must equal the number of unknown quantities • Check – Consider what your answer means – Check that your answer makes sense Inclines (Hills) • Normal force (N) : perpendicular to the incline (plane) • Friction force : up the incline – Opposite to motion • The force due to gravity acts straight down • Coordinate system – Axes parallel and perpendicular to the incline • (If) Acceleration : along the incline • Components of the gravitational force • The normal force is not equal to mg Angle of Incline To Not Slip • The minimum frictional force to keep the object from slipping is • Ffriction = m g sin θ • Since this is static friction, Ffriction ≤ μstatic N • Assuming it is just in equilibrium (so Ffriction = μstatic N), the angle of the incline at which the object is on the verge of slipping is tan θ = μs Equilibrium Example, Flag • Find out tension and angle (two unknowns so 2D problem) • Free body diagram – Horizontal and vertical directions : coordinate system – Tension has x- and ycomponents • Equations for equilibrium in the x- and y-directions Motion in Two Dimensions • Two dimensions is just like one dimension, done twice: • for the x components, then • for the y components. • The x and y motions are independent of each other! • The x part of the motion occurs exactly as if the y part did not exist. • And, the y part of the motion occurs exactly as if the x part did not exist. Projectile Motion The x and y motions are independent of each other! In general, for the x components: assume that there is no air resistance, so that ax = 0 then vx = constant = v0x In general, for the y components: ay = 9.8 m/s2 down then vy changes Acceleration due to gravity is always 9.8 m/s2 downward, throughout the path of the projectile! y v θ x Projectile Motion • Consider objects in motion and the forces acting on them • Projectile motion is one example of this type of motion • We will ignore the force from air drag – For now • Components of gravity are Fgrav, x = 0, Fgrav, y = - m g HW Question Calculate the distance between the window and the ceiling using kinematics A roofing tile falls from rest off the roof of a building. An observer from across the street notices that it takes 0.54 s for the tile to pass between two windowsills that are 2.5 m apart. How far is the sill of the upper window from the roof of the building? The Three Equations (1) v v0 at (2) 1 2 x v0 t at 2 (3) v v 2ax 2 2 0 Tomorrow: (ch 4) Projectile Motion Reference Frames & Relative Velocity Example Involving Newton’s Law