# Lecture15 forces

YogaTrail
4 Jan 2012
1 sur 27

### Lecture15 forces

• 1. Force and Motion Girl pulls table cloth quickly but dishes remain on the table – a lot of physics is happening here
• 2. Inertia According to Aristotle, the natural state of objects was to be at rest, and if you got them moving, eventually they would come to rest again. But being at rest is relative . Galileo did experiments rolling balls down and up inclined planes, and realized that, in the absence of some kind of force, an object would keep moving forever once it got started. Galileo called this phenomenon inertia
• 3. Inertial Frames If you’re on a train moving 10 m/s, the world outside seems to be moving very fast. But if you look at objects inside the train, everything is still. So, objects are at rest only in some frames. If a frame is not accelerating, it’s called an inertial frame. Root word: inert If there are two trains, and the ground, there are many relative velocities. Velocities are relative! Consider that the earth is turning too…
• 4. Inertia Inertia is the natural tendency of an object to maintain a state of rest or to remain in uniform motion in a straight line (constant velocity) in an inertial frame. Mass is a measure of how much inertia an object has. Massive objects have more inertia, and it is more difficult to change the motion of a massive object.
• 5. Newton ’s Laws of Motion Newton ’s 1 st law of motion is sometimes called the law of inertia: In the absence of any outside forces, a body at rest remains at rest, and a body already in motion remains in motion with a constant velocity (constant speed and direction).
• 6. Force and Net Force A force is something that is capable of changing an object ’s velocity. Any particular force may not actually change an object ’s velocity, because there may be other forces that prevent it from doing so. But if the net force – the vector sum of all forces acting on the object – is not zero, the velocity of the object will change.
• 7. Force and Net Force
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• 9. Newton ’s 2 nd Law of Motion Experiments show that the acceleration of an object is proportional to the force exerted on it and inversely proportional to its mass. The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of the acceleration is in the direction of the applied net force. F = m a
• 10. Newton ’s 2 nd Law of Motion The units of force are called Newtons. 1 N = 1 kg . m/s 2 .
• 11. Newton ’s 2 nd Law of Motion An object ’s weight is the force exerted on it by gravity. Here, g is the acceleration of gravity: g = 9.81 m/s 2 Weight therefore has the same units as force – Newtons
• 12. Newton ’s 2 nd Law of Motion Newton ’s second law applies separately to each component of the force:
• 13. Newton ’s 3 rd Law of Motion For every force (action), there is an equal and opposite force (reaction). Note that the action and reaction forces act on different objects. This image shows how a block exerts a downward force on a table; the table exerts an equal and opposite force on the block, called the normal force N .
• 14. Newton ’s 3 rd Law of Motion This figure illustrates the action-reaction forces for a person carrying a briefcase. Is there a reaction force in (b)? If so, what is it?
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• 16. Free-Body Diagrams A free-body diagram draws the forces on an object as though they all act at a given point. You should draw such a diagram whenever you are solving problems.
• 17. Free-Body Diagrams If an object has no acceleration, there must be no net force on it. Or, said in a different way, when there is no net force on an object, its acceleration is 0 This translates into three separate requirements: – that there be no force in the x direction, the y direction, or the z direction. Always draw a free body diagram with all the forces acting on an object in order to analyze the situation
• 18. Free-Body Diagram: Simple Example Ball hanging on a string Draw all Forces acting on ball In x direction? In y direction? There is no NET force on the ball, so there is no acceleration mg T
• 19. Free-Body Diagrams Example: pendulum from our first Lab: Drawing free-body diagram: there are two forces on the ball: Tension (T) and gravity (mg). When the angle  is not 0, is there a net force acting on the ball? What about when  is 0?
• 20. Free-Body Diagrams Example: pendulum from our first Lab: y’ x’ First, change coordinate system to make analysis easier Often, choosing coordinates where some of the forces are along coordinate axis simplifies the math of a problem considerably…
• 21. Free-Body Diagrams Example: pendulum from our first Lab: Change coordinate system to make analysis easier In y’ direction, ball cannot move, so acceleration in y’ is 0: y’ x’ mg cos(  ) F y = T - mg cos(  ) = 0 mg sin(  ) F x = mg sin(  ) ≠ 0
• 22. Free-Body Diagrams Example: pendulum from our first Lab: So in the picture, there is an acceleration along x’, and the ball moves in that direction F x = mg sin(  ) = ma  a = g sin(  ) y’ x’ mg cos(  ) mg sin(  )
• 23. Free-Body Diagrams When the mass/string system is directly vertical, there is still an acceleration acting on the mass. Why? In what direction? If the mass starts from θ = 10°, and the string length is L, can we calculate the maximum speed it has (when passing through the bottom of the motion?) Yes, consider energy: ΔU = mgΔh, at the bottom the KE = ½ m v 2 , so
• 24. Free-Body Diagrams so Δh = L(1 - cos [  ] ) y’ x’
• 25. Period of a pendulum We learned the equation of motion in x and y are completely separate. Consider a “conical pendulum”: it can be arranged to provide perfectly circular motion for a mass, where the mass never changes its vertical level
• 26. Period of a pendulum Circular motion is described by sines and cosines. If we look at the thing from the side, we see the mass going from left to right and back again The mass goes around the circle with v max , and the period is just the time it takes to go around: T = 2πR/v max R = L sin[  ]
• 27. Period of a pendulum The period is Here an approximation was made: when the angle  is small, the term involving the angle is very nearly equal to 1….