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## Physics

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**Physics**Chapter 7 Forces and Motion in Two Dimensions**Forces and Motion in Two Dimensions**• 7.1 Forces in Two Dimensions • 7.2 Projectile Motion • 7.3 Circular Motion • And…Simple Harmonic Motion from Chapter 6!**Forces in Two Dimensions**• We have already studied a few forces in two dimensions • When dealing with friction acting on an object (parallel), the normal force on the object (perpendicular) also plays a role in friction • In this chapter we will be looking at forces on a object that are at angles other than 90°**Equilibrium and the Equilibrant**• Equilibrium—when the net forces on an object is equal to zero • This means an object can be stationary or moving but it can’t be _________. • When graphically adding vectors together, if a closed geometric shape is formed, there is no resultant (?) so net force is zero**Equilibrium and the Equilibrant**If an object is not in equilibrium this means that: • There is a net force on it • When the vectors are added together, there is a resultant • It is accelerating**Equilibrium and the Equilibrant**• If add a vector to the resultant that is equal and opposite to it, there will be no net force and the object will be in equilibrium • This vector is called the equilibrant • The equilibrant is the vector that puts an object in equilibrium • It is equal and opposite to the resultant of the existing vectors**Creating Equilibrium**Hanging a Sign • A 168 N sign is equally supported by two ropes that make an angle between them of 135°. What is the tension in the rope? 168 N**Creating Equilibrium**Hanging a Sign • Is 168 N the sign’s weight, resultant force, and/or equilibrant? 168 N**Creating Equilibrium**Hanging a Sign • So both ropes together must pull up with a force of 168 N to put the sign into equilibrium 168 N**Creating Equilibrium**Hanging a Sign • So each ropes together must pull up with a force of 84 N (1682) to put the sign into equilibrium 168 N 84 N**Creating Equilibrium**Hanging a Sign • So to find the tension in the rope (hypotenuse of a right triangle) use trig! • cos = a/h so h = a/cos • H = 84/cos 67.5 • H = 219.5 N • Why was = 67.5° and not 135°? 84 N**Creating Equilibrium**Try this….. Hanging a Sign • A 150 N sign is equally supported by two ropes that make an angle between them of 96°. What is the tension in the rope? • Answer: • 112 N**Motion Along an Inclined Plane**• When an object sits on a flat surface, there are four forces that determine if there is a net force acting on the object. • What are these forces? (five force equation)**Motion Along an Inclined Plane**• If the object is in equilibrium, what can you say about the relationship between Fg, Fn, Fa and Ff?**Motion Along an Inclined Plane**• If the object is in equilibrium on a sloping surface, which of the following forces doesn’t change; Fg, Fn, Fa and Ff?**Motion Along an Inclined Plane**• If the object is in equilibrium on a sloping surface, which of the following forces doesn’t depend on the angle of the slope; Fg, Fn, Fa and Ff?**Motion Along an Inclined Plane**• The object weight (Fg) is always the same (regardless of the slope of the surface) and it always directed straight down (toward center of the earth).**Motion Along an Inclined Plane**• Fn, Fa, Ff all change with the angle of the slope**Motion Along an Inclined Plane**• So lets look how Fg, Fa, Fn, and Ff are related**Motion Along an Inclined Plane**• When an object sits on an inclined plane it is being pulled into the plane and down the plane by the force of gravity Fg**Motion Along an Inclined Plane**• Fg is always the hypotenuse of the right force triangle formed Fg**Motion Along an Inclined Plane**• The force with which the object is pulled into the plane is called the perpendicular force (F) F Fg**Motion Along an Inclined Plane**• The force with which the object is pulleddown the plane is called the parallel force (F) F Fg F**Motion Along an Inclined Plane**• These forces are perpendicular and parallel to what? F Fg F**Motion Along an Inclined Plane**• We could solve for F and F|| using Fg and angle if we knew what was. • Angle is always equal to the slope of the plane F Fg F**Motion Along an Inclined Plane**• So for F; cos = a/h • Since a = F and h = Fg • F = Fg cos F = Fg cos Fg F**Motion Along an Inclined Plane**• So for F; sin = o/h • Since o = F and h = Fg • F = Fg sin F = Fg cos Fg F = Fg sin **Motion Along an Inclined Plane**• So if gravity pulls the object down into the plane with a force F , which force counteracts it? F = Fg cos Fg F = Fg sin **Motion Along an Inclined Plane**• So F , is equal and opposite Fn Fn F = Fg cos Fg F = Fg sin **Motion Along an Inclined Plane**• So if gravity pulls the object down the plane with a force F ,which force counteracts it? F = Fg cos Fg F = Fg sin **Motion Along an Inclined Plane**• So F is equal and opposite to Ff • If there is no motion (or acceleration)! Ff F = Fg cos Fg F = Fg sin **Motion Along an Inclined Plane**• So F , is equal and opposite Fn • and F is equal and opposite to Ff • Only if there is no ________or_______.**Motion Along an Inclined Plane**Example • A trunk weighing 562 N is resting on a plane inclined 30°. Find the perpendicular and parallel components of its weight. • Answers F = - 487 N F = - 281 N • Why are they negative? • What other forces are they equal to? • Would the force change if the object was moving?**Motion Along an Inclined Plane**• Under what set of circumstances could the object be accelerating down the plane? • F > Ff • Slope too slippery—not enough friction to hold it • We could be pushing or pulling the object • What can we say about the net force on the object?**Motion Along an Inclined Plane**What is the Five Forces Equation? • Fnet = Fa + Ff + Fg + Fn • Lets see how it can be modified to deal with inclined planes**Motion Along an Inclined Plane**Fnet = Fa + Ff + Fg + Fn • Does Fn determine if the object will accelerate down the plane? • So we get rid of it • Fnet = Fa + Ff + Fg**Motion Along an Inclined Plane**Fnet = Fa + Ff + Fg • What are the two components of weight for an object on the plane? • We replace weight with these two • Fnet = Fa + Ff + F + F**Motion Along an Inclined Plane**Fnet = Fa + Ff + F + F • Does F determine if the object accelerates down the plane? • Get rid of it • Fnet = Fa + Ff+ F**Motion Along an Inclined Plane**Fnet = Fa + Ff+ F • Can we still push or pull the object? • Does friction still act on it? • So this is the new Four Forces Equation for inclined planes Fnet = Fa + Ff+ F**Motion Along an Inclined Plane**Example • A trunk weighing 562 N is resting on a plane inclined 30°. Find the perpendicular and parallel components of its weight. If the force of friction is 200 N, find the trunk’s acceleration rate. • Answers F = - 487 N F = - 281 N a = -1.41 m/s2**Projectile Motion**Projectile—a launched object that moves through air only under the force of gravity • Ignoring air resistance! Trajectory—the path of a projectile through space**Projectile Motion**A projectile has both horizontal and vertical components of its velocity • These components are independent of each other • This is because the force of gravity acts on the vertical component (causing acceleration) but not on the horizontal component (constant velocity). • But one can equal zero!**Projectile Motion**We will be learning how to solve three types of projectile motion problems: • Dropped Objects • Objects Thrown Horizontally • Objects Launched at an Angle**A Dropped Object**• When an object is dropped from a height it will fall • It has two components of its velocity, horizontal and vertical and both are equal to zero**A Dropped Object**• As it falls, it picks up speed (accelerates) in the vertical direction due to the force of gravity • Acceleration due to gravity =? • So when dropped any object will pick up negative vertical velocity at the rate of – 9.8 m/s for each second it falls**A Dropped Object**• As it falls, its horizontal speed stays constant (why?) • So as an object falls its vertical speed changes but its horizontal speed doesn’t • These two perpendicular components of speed are independent of each other, they have no effect on each other (this is true of all perpendicular vectors)**A Dropped Object**Example • A 2.5 kg stone is dropped from a cliff 44 m high. How long is it in the air? • Answer 3.0 sec • What is its velocity right before it hits the ground? • Answer - 29.4 m/s**An Object Thrown Horizontally**• When an object is thrown horizontally, it has horizontal and vertical components to its velocity • The horizontal velocity is constant during the entire time its in the air (why?) • The vertical velocity starts as zero but increases as it falls just as if it were dropped! (why?) • These two components are independent of each other**An Object Thrown Horizontally**Example • A 2.5 kg stone thrown horizontally at 15 m/s from a cliff 44 m high. • How long is it in the air? • What is its horizontal velocity right before it hits the ground? • What is its vertical velocity right before it hits the ground? • How far from the cliff does it land?**An Object Thrown Horizontally**Example • A 2.5 kg stone thrown horizontally at 15 m/s from a cliff 44 m high. • How long is it in the air? • Answer 3.0 sec