Motion and Forces in Two Dimensions Sec. 7.1 Forces in Two Dimensions 1. A Review of Vector Addition. Forces on an Inclined Plane 3. How to find an Equilibrant Vector 4. Projectile Motion Objectives Determine the force that produces equilibrium when many forces act on an object Analyze the motion of an object on an inclined plane with and without friction A Review of Vector Addition Weight Forces on an Inclined Plane Direction Matters! The process of breaking a vector into its x and y components is called VECTOR RESOLUTION. We can use it to add vectors together. Here s how we can use vector resolution to add vectors A and B to find the magnitude and direction of the resultant vector, C. We break A into its components, A x and A y. A x = A cos θ A A y = A sin θ A We break B into its components, B x and B y. B x = B cos θ B B y = B sin θ B 1
We then add the x- components: A x + B x = C x (or R x ) And we add the y- components: A y + B y = C y (or R y ) We then can determine the magnitude of vector C, by adding together its components (vector resolution in reverse): C x + C y = C or, aka R x + R y = R Of course, we are adding perpendicular vectors here, so we must use the Pythagorean Theorem: R = R x + R y What about the direction of vector C? Look at the purple triangle: we now have a right angle, formed from C x and C y. We can determine the direction with some right angle trig: (tan -1 ). Traps and Pitfalls: Be aware of how you define θ; it may not be counterclockwise from east! Be aware of signs (+ and -), especially when adding vectors in different quadrants! θ = tan -1 (C y /C x ) Components of Vectors So far, we have usually dealt with vectors in the first quadrant, and θ has been measured from the x axis. In that case, A = (A x + A y ) A x =A cos θ A y = A sin θ A A x A y But soon we will need to be proficient with applying trig functions to different situations θ Why are we reviewing this? A projectile s instantaneous velocity can be resolved into horizontal and vertical components.
Let s : Let s : eastward with 1. N, and a second force pulls south at 14.0 N. eastward with 4.0 N, and a second force pulls 55 degrees north of east with 18.0 N. Let s : eastward with 4.0 N, and a second force pulls due northwest with 18.0 N. Let s : Newton s Laws and Friction Review: The coefficient of kinetic friction between a 15 kg crate and a concrete floor is 0.1. What force must be applied to make the crate move at a constant speed of 1. m/s? Forces and Motion on an Inclined Plane Inclined Plane Problems: Strategies Draw a free body diagram. Choose a coordinate system: Make plane s surface be x axis. Make the y axis be perpendicular to the plane. 3
Inclined Plane Problems: Strategies The inclined plane exerts an upward force perpendicular to its surface, this is the NORMAL FORCE. It is 90 o to the surface, not 90 o to the weight force! NOTE: F N is not equal and opposite to F g! Inclined Plane Problems: Strategies Since the box has no acceleration in the y- direction all forces in that direction must balance. Therefore we get the following equations: F N + F gy = 0 F N = - F gy F gy = F g cos F N = - F g cos What is the magnitude of F N? F N A 55-kg box is at rest on a 6 o inclined plane. What is the normal force on the box? A 55-kg box is at rest on a 6 o inclined plane. What is the net force on the box? A 55-kg box is sliding at constant velocity on a 6 o inclined plane. What is the net force on the box? A 55-kg box is at placed on a frictionless surface which is a 6 o inclined plane. What is the net force on the box? acceleration =? 4
A 75-kg skier is sliding down a 16 o inclined plane at constant speed. What is the friction force between the slope on the skis? Important Concepts and Vocabulary Resultant Force vector sum of or more vectors. Equilibrium condition in which net force on an object is zero. When the net force is zero the object is in equilibrium. Equilibrant Force The force needed to bring an object into equilibrium. Force that is applied to produce equilibrium. We will use this for the lab. It is the single additional force that if applied to the same point as the other forces, will produce equilibrium. To find the equilibrant find the Resultant Force. The equilibrant force is equal in magnitude to the resultant but opposite in direction. So add 180. Equilibrant We know that for an object to be at rest, F net must = 0. How to find an Equilibrant Vector We often must ask, What force additional force must be applied to make F net must = 0? This force, the force exerted on an object to produce equilibrium, is called the equilibrant. It has the same magnitude as the resultant or net force, but it is opposite in direction Finding the Equilibrant Force Find the net force, or the resultant force. (Use vector addition) Finding the Equilibrant Force Try it: A 1.0 N force is applied at 0.00 o (east) to a 15.0 N object. What would the equilibrant be? Note: this object is not on a surface! Change the direction by 180 o! F R = 19. N; 308.7 F E = 19. N; 18.7 Our strategy: Find the net force, or the resultant force. Change the direction 180 o! 5