1. Chapter 5 Force and Motion I

2. Classical Mechanics Describes the relationship between the motion of objects in our everyday world and the forces acting on them Describes the relationship between the motion of objects in our everyday world and the forces acting on them Conditions when Classical Mechanics does not apply Conditions when Classical Mechanics does not apply very tiny objects (< atomic sizes) very tiny objects (< atomic sizes) objects moving near the speed of light objects moving near the speed of light

3. Forces Usually think of a force as a push or pull Usually think of a force as a push or pull Vector quantity Vector quantity May be contact or field force May be contact or field force

4. Contact and Field Forces

5. Fundamental Forces Types Types Strong nuclear force Strong nuclear force Electromagnetic force Electromagnetic force Weak nuclear force Weak nuclear force Gravity Gravity Characteristics Characteristics All field forces All field forces Listed in order of decreasing strength Listed in order of decreasing strength Only gravity and electromagnetic in mechanics Only gravity and electromagnetic in mechanics

6. Measuring the Strength of a Force Use the deformation of a spring to determine the strength of F. Use the deformation of a spring to determine the strength of F. The spring elongates under the action of F 1 and the pointer on the scale indicates the magnitude of F 1. The spring elongates under the action of F 1 and the pointer on the scale indicates the magnitude of F 1. Calibration is necessary Calibration is necessary

7. Newton’s First Law If no forces act on an object, it continues in its original state of motion; that is, unless something exerts an external force on it, an object at rest remains at rest and an object moving with some velocity continues with that same velocity. If no forces act on an object, it continues in its original state of motion; that is, unless something exerts an external force on it, an object at rest remains at rest and an object moving with some velocity continues with that same velocity.

8. Newton’s First Law, cont. External force External force any force that results from the interaction between the object and its environment any force that results from the interaction between the object and its environment Alternative statement of Newton’s First Law Alternative statement of Newton’s First Law When there are no external forces acting on an object, the acceleration of the object is zero. When there are no external forces acting on an object, the acceleration of the object is zero.

9. Inertia Is the tendency of an object to continue in its original motion Is the tendency of an object to continue in its original motion

10. Mass A measure of the resistance of an object to changes in its motion due to a force A measure of the resistance of an object to changes in its motion due to a force Scalar quantity Scalar quantity SI units are kg SI units are kg

11. Newton’s Second Law The acceleration of an object is directly proportional to the net force acting 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. F and a are both vectors F and a are both vectors Can also be applied three-dimensionally Can also be applied three-dimensionally

12. Units of Force SI unit of force is a Newton (N) SI unit of force is a Newton (N) US Customary unit of force is a pound (lb) N. B. (1 N = 0.225 lb) US Customary unit of force is a pound (lb) N. B. (1 N = 0.225 lb)

13. A sphere and a block are connected by a string placed over a pulley. The two objects are at rest at the same level, as shown in the Figure. A teacher performs the following demonstration: The block is now pulled down and held while in its new position, as shown in Figure 2. The teacher asks the class to predict what will happen in this real situation when the block is released, and to give a reason supporting the prediction.

14. A teacher performs the following demonstration: 1.The block will move down because the gravitational force on the block will be greater at this lower level. 2. The block will remain stationary because there will be no resultant force on the block. 3. The block will move up and return to its original position because this will conserve potential energy. 4. The block will move up and return to its original position because that was its equilibrium position.

15. More Question: The adjacent figure shows overhead views of four situations in which forces act on a block that lies on a frictionless floor. If the force magnitudes are chosen properly, in which situations is it possible that the block is (a) stationary and (b) moving with a constant velocity?

16. Gravitational Force Mutual force of attraction between any two objects Mutual force of attraction between any two objects Expressed by Newton’s Law of Universal Gravitation: Expressed by Newton’s Law of Universal Gravitation:

17. Weight The magnitude of the gravitational force acting on an object of mass m near the Earth’s surface is called the weight w of the object The magnitude of the gravitational force acting on an object of mass m near the Earth’s surface is called the weight w of the object w = m g is a special case of Newton’s Second Law w = m g is a special case of Newton’s Second Law g can also be found from the Law of Universal Gravitation g can also be found from the Law of Universal Gravitation

18. More about weight Weight is not an inherent property of an object Weight is not an inherent property of an object mass is an inherent property mass is an inherent property Weight depends upon location Weight depends upon location

19. Newton’s Third Law If two objects interact, the force F 12 exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force F 21 exerted by object 2 on object 1. If two objects interact, the force F 12 exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force F 21 exerted by object 2 on object 1. Equivalent to saying a single isolated force cannot exist Equivalent to saying a single isolated force cannot exist

20. Newton’s Third Law cont. F 12 may be called the action force and F 21 the reaction force F 12 may be called the action force and F 21 the reaction force Actually, either force can be the action or the reaction force Actually, either force can be the action or the reaction force The action and reaction forces act on different objects The action and reaction forces act on different objects

21. Some Action-Reaction Pairs n and n’ n and n’ n is the normal force, the force the table exerts on the TV n is the normal force, the force the table exerts on the TV n is always perpendicular to the surface n is always perpendicular to the surface n’ is the reaction – the TV on the table n’ is the reaction – the TV on the table n = - n’ n = - n’

22. More Action-Reaction pairs F g and F g ’ F g and F g ’ F g is the force the Earth exerts on the object F g is the force the Earth exerts on the object F g ’ is the force the object exerts on the earth F g ’ is the force the object exerts on the earth F g = -F g ’ F g = -F g ’

23. Forces Acting on an Object Newton’s Law uses the forces acting on an object Newton’s Law uses the forces acting on an object n and F g are acting on the object n and F g are acting on the object n’ and F g ’ are acting on other objects n’ and F g ’ are acting on other objects

24. The adjacent Figure shows four choices for the direction of a force of magnitude F to be applied to a block on an inclined plane. The directions are either horizontal or vertical. (For choices a and b, the force is not enough to lift the block off the plane.) Rank the choices according to the magnitude of the normal force on the block from the plane, greatest first. Forces Acting on an Object Ex.:

25. Applying Newton’s Laws Assumptions Assumptions Objects behave as particles Objects behave as particles can ignore rotational motion (for now) can ignore rotational motion (for now) Masses of strings or ropes are negligible Masses of strings or ropes are negligible Interested only in the forces acting on the object Interested only in the forces acting on the object

26. Free Body Diagram Must identify all the forces acting on the object of interest Must identify all the forces acting on the object of interest Choose an appropriate coordinate system Choose an appropriate coordinate system If the free body diagram is incorrect, the solution will likely be incorrect If the free body diagram is incorrect, the solution will likely be incorrect

27. Equilibrium An object either at rest or moving with a constant velocity is said to be in equilibrium An object either at rest or moving with a constant velocity is said to be in equilibrium The net force acting on the object is zero The net force acting on the object is zero

28. Equilibrium cont. Easier to work with the equation in terms of its components: Easier to work with the equation in terms of its components:

29. Solving Equilibrium Problems Make a sketch of the situation described in the problem Make a sketch of the situation described in the problem Draw a free body diagram for the isolated object under consideration and label all the forces acting on it Draw a free body diagram for the isolated object under consideration and label all the forces acting on it Resolve the forces into x- and y-components, using a convenient coordinate system Resolve the forces into x- and y-components, using a convenient coordinate system Apply equations, keeping track of signs Apply equations, keeping track of signs Solve the resulting equations Solve the resulting equations

30. Equilibrium Example – Free Body Diagrams

31. Newton’s Second Law Problems Similar to equilibrium except Similar to equilibrium except Use components Use components a x or a y may be zero a x or a y may be zero

32. Solving Newton’s Second Law Problems Make a sketch of the situation described in the problem Make a sketch of the situation described in the problem Draw a free body diagram for the isolated object under consideration and label all the forces acting on it Draw a free body diagram for the isolated object under consideration and label all the forces acting on it If more than one object is present, draw free body diagram for each object If more than one object is present, draw free body diagram for each object Resolve the forces into x- and y-components, using a convenient coordinate system Resolve the forces into x- and y-components, using a convenient coordinate system Apply equations, keeping track of signs Apply equations, keeping track of signs Solve the resulting equations Solve the resulting equations

33. Inclined Planes Choose the coordinate system with x along the incline and y perpendicular to the incline Choose the coordinate system with x along the incline and y perpendicular to the incline Replace the force of gravity with its components Replace the force of gravity with its components

34. Example: A car of mass m is on a icy driveway inclined at angle  as shown in the adjacent Figure. A car of mass m is on a icy driveway inclined at angle  as shown in the adjacent Figure. Find the acceleration of the car, assuming No Friction.

35. Solution: Chose of x- and y-axes. Chose of x- and y-axes. Free Body diagram. Free Body diagram. Newton’s 2 nd Law: Newton’s 2 nd Law: mg + n = m a mg + n = m a Projection along the x-axis: mgsin(  ) = m a x mgsin(  ) = m a x Projection along the y-axis: mgcos(  ) + n = 0 mgcos(  ) + n = 0  a x = g  sin(  )

36. More Examples: Two Blocks of masses m 1 and m 2, with m 1 > m 2, are placed in contact with each other on a frictionless horizontal surface. A constant force F is applied to m 1 as shown. Two Blocks of masses m 1 and m 2, with m 1 > m 2, are placed in contact with each other on a frictionless horizontal surface. A constant force F is applied to m 1 as shown. a) Find the magnitude of the acceleration of the system? b) Determine the magnitude of the contact force between the two blocks?

37. Solution: Both Blocks must experience the same acceleration since they are in contact. Both Blocks must experience the same acceleration since they are in contact. F is the only force acting horizontally: F is the only force acting horizontally: F = (m 1 +m 2 ) a x

38. Solution Cont’d: Free body diagram of m 1. Free body diagram of m 1. Along the x-axis, we have: F – P 21 = m 1 a x x y

39. Solution: Note that one can get P 12 from m 2 : Note that one can get P 12 from m 2 : Along the x-axis: P 12 = m 2 a x P 12 = m 2 a x We see that |P 12 |= |P 21 |

40. Example, The Atwood Machine: Two object of unequal masses (m 2 >m 1 ) are hung vertically over a frictionless pulley of negligible mass. Two object of unequal masses (m 2 >m 1 ) are hung vertically over a frictionless pulley of negligible mass. Determine the magnitude of the acceleration of the two objects and the tension in the lightweight cord.

41. Solution Atwood Machine: m 2 will move down and m 1 will move up with the same a. m 2 will move down and m 1 will move up with the same a. N.B. The tensions in the cord from both side are equal since mass of pulley is negligible and there is no Friction. N.B. The tensions in the cord from both side are equal since mass of pulley is negligible and there is no Friction. a a

42. Atwood Cont’d: a Drawing the free body diagram and Drawing the free body diagram and applying Newton’s second Law. applying Newton’s second Law. Projection along the direction of motion (upwards for m 1 ) -m 1 g + T = m 1 a  T = m 1 g + m 1 a Projection along the direction of motion (downwards for m 2 ) m 2 g - T = m 2 a  T = m 2 g – m 2 a

43. Atwood Cont’d: T = T  m 1 g + m 1 a = m 2 g – m 2 a (m 2 – m 1 )g = Net Force (m 2 + m 1 ) = Total mass N.B.