Forces and Motion in Two Dimensions

1. Chapter 7 Forces and Motion in Two Dimensions

2. 7.1 Forces in Two Dimensions An object is in equilibrium when the net force on it is zero. When in equilibrium the object is motionless or move with constant velocity. Equilibrium also occurs when the resultant of three or more forces = a net force of zero. The equilibrant force is equal in magnitude but opposite in direction to the resultant vector.

3. Example Problem Creating Equilibrium A 168-N sign is supported in a motionless position by two ropes that each make 22.5? angles with the horizontal. What is the tension in the ropes?

4. Motion along an incline Plane A tilted surface is called an inclined plane. Objects accelerate down inclined planes because of an unbalanced force. The two forces acting upon a crate which is positioned on an inclined plane (assumed to be friction-free) are the force of gravity and the normal force. The force of gravity (weight) acts in a downward direction; yet the normal force acts in a direction perpendicular to the surface.

6. Example Problem Components of Weight for an Object on an incline. A trunk weighing 562 N is resting on a plane inclined 30.0? above the horizontal. Find the components of the weight force parallel and perpendicular to the plane. Skiing Downhill A 62-kg person on skis is going down a hill sloped at 37 ?. The coefficient of kinetic friction between the skis and the snow is 0.15. How fast is the skier going 5.0 s after starting from rest?

7. 7.2 Projectile Motion A projectile can be a football, a bullet, or a drop of water. A projectile is any object which once projected continues in motion by its own inertia and is influenced only by the downward force of gravity.

9. At a given location on the earth and in the absence of air resistance, all objects fall with the same uniform acceleration. Thus, two objects of different sizes and weights, dropped from the same height, will hit the ground at the same time.

10. An object is controlled by two independent motions. So an object projected horizontally will reach the ground at the same time as an object dropped vertically. No matter how large the horizontal velocity is, the downward pull of gravity is always the same.

11. The horizontal motion of the cannonball is the result of its own inertia.As the cannonball falls, it undergoes a downward acceleration.

12. The Independence of the horizontal and vertical directions means that a projectile motion problem consists of two independent parts: Vertical motion at a constant downward acceleration, which is equal to a = -g = -9.80 m/s2. Horizontal motion at a constant horizontal speed, vx= constant.

13. The object's vertical motion is the same as that of an object undergoing only vertical free-fall. Gravity only affects the object's vertical motion. Gravity cannot change the object's horizontal speed, and the component of the object's horizontal velocity remains constant throughout its motion.

14. Summary of Projectile Motion Equations

15. Example Problem A Projectile Launched Horizontally A stone is thrown horizontally at 15 m/s from the top of a cliff 44 m high.. how far from the base of the cliff does the stone hit the ground? How fast is it moving the instant before it hits the ground? (Check this site for more fun problems: http://www.crocodile-clips.com/absorb/AP5/sample/010105.html)

16. projectile is launched upward at an angle to the horizontal

17. When a projectile is launched at an angle, the initial velocity has a vertical component as well as a horizontal component. Maximum height: is the height of the projectile when the vertical velocity is zero and the projectile has only its horizontal velocity component. Range, R: is the horizontal distance the projectile travels. Flight time: is the time the projectile is in the air. (called hang time in football game)

18. Example Problem The Flight of a Ball The ball in the photo was launched with an initial velocity of 4.47 m/s at an angle of 66? above the horizontal. a. What was the maximum height the ball attained? b. How long did it take the ball to return to the launching height? c. What was its range?

19. 7.3 Circular Motion Uniform circular motion is the motion of an object in a circle with a constant or uniform speed. As an object moves in a circle, it is accelerating inward due to its change in direction.

21. Centripetal Acceleration

22. Speed of an object moving in circle is: v = 2?r /T T, is the period, which is the time needed for the object to make a complete revolution. During this time it travels a distance= to the circumference of the circle, 2?r.

23. Newton’s Second Law Fnet = m×ac

24. Example Problem Uniform Circular Motion A 13-g bucket is attached to a 0.93-m string. The bucket is swung in a horizontal circle, making one revolution in 1.18 s. Find the tension force exerted by the string on the bucket.

26. The equation for centripetal acceleration is:ac = v2 / r, Where ac is centripetal acceleration, v is velocity in meters per second, and r is the radius of the circle in meters. 

27. Roller Coaster G-Forces

28. G-Forces  G-forces are used for explaining the relative effects of centripetal acceleration that a rider feels while on a roller coaster. The greater the centripetal acceleration, the greater the G-forces felt by the passengers. A force of 1 G is the usual force of the Earth’s gravitational pull that a person feels when they are at rest on the Earth’s surface; it is a person’s normal weight. When a person feels weightless, as in free fall or in space, they are experiencing 0 G’s. When the roller coaster train is going down a hill, the passengers usually undergo somewhere between 0 and 1 G. If the top of the hill is curved more narrowly than a parabola, the passengers will experience negative G’s as they rise above the seat and get pushed down by the lap bar. This is because gravity and the passengers’ inertia would have them fall in a parabolic arc. G-forces greater than 1 can be felt at the bottom of hills as the train changes direction. In this case the train is pushing up on the passengers with more than the force of gravity because it is changing their direction of movement from down to up. G-forces that are felt when changing direction horizontally are called lateral G’s. Lateral G’s can be converted into normal G-forces by banking turns.

29. Inertia and the Right Hand Turn

30. If all forces on the car ceased at point A, it would continue along a straight line to point B, in accordance with Newton's first law. If only gravity acted, it would follow a parabola to point C. For the rails to exert a positive pressure, they must constrain the car to a tighter curvature than gravity alone, forcing it to move to point D.