Ying Yi PhD Chapter 4 The Laws of Motion 1 PHYS HCC

Outline PHYS HCC 2 Force Newton’s Three Law Force example 1: Gravitational force Force example 2: Friction Application of Newton’s Laws

PHYS HCC 3 Sir Isaac Newton 1642 – 1727 Formulated basic concepts and laws of mechanics Universal Gravitation Calculus Light and optics

Classical Mechanics 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 Very tiny objects (< atomic sizes) Objects moving near the speed of light 4 PHYS HCC

Contact and Field Forces 5 PHYS HCC

Fundamental Forces Types Strong nuclear force Electromagnetic force Weak nuclear force Gravity Characteristics All field forces Listed in order of decreasing strength Only gravity and electromagnetic in mechanics 6 PHYS HCC

Newton’s First Law An object moves with a velocity that is constant in magnitude and direction, unless acted on by a nonzero net force Note that: The net force is defined as the vector sum of all the external forces exerted on the object 7 PHYS HCC

External and Internal Forces External force Any force that results from the interaction between the object and its environment Internal forces Forces that originate within the object itself They cannot change the object’s velocity 8 PHYS HCC

Inertia Is the tendency of an object to continue in its original motion In the absence of a force Thought experiment Hit a golf ball Hit a bowling ball with the same force The golf ball will travel farther Both resist changes in their motion 9 PHYS HCC

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. Can also be applied three-dimensionally 10 PHYS HCC

Units of Force SI unit of force is a Newton (N) US Customary unit of force is a pound (lb) 1 N = lb See table PHYS HCC

Some Notes About Forces Forces cause changes in motion Motion can occur in the absence of forces All the forces acting on an object are added as vectors to find the net force acting on the object m is not a force itself Newton’s Second Law is a vector equation 12 PHYS HCC

Example 4.2 Horses Pulling a Barge PHYS HCC 13 Two horses are pulling a barge with mass 2.00×10 3 Kg along a canal, as shown in Figure 4.7. The cable connected to the first horse makes an angle of Ɵ 1 =30.0° with respect to the direction of the canal, while the cable connected to the second horse makes an angle of Ɵ 2 =-45.0°. Find the initial acceleration of the barge, starting at rest, if each horse exerts a force of magnitude 6.00×10 2 N on the barge. Ignore forces of resistance on the barge.

Newton’s Third Law If object 1 and object 2 interact, the force exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force exerted by object 2 on object 1. Equivalent to saying a single isolated force cannot exist 14 PHYS HCC

15 Newton’s Third Law cont. 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 The action and reaction forces act on different objects

PHYS HCC 16 Some Action-Reaction Pairs is the normal force, the force the table exerts on the TV is always perpendicular to the surface is the reaction – the TV on the table

PHYS HCC 17 More Action-Reaction pairs is the force the Earth exerts on the object is the force the object exerts on the earth

PHYS HCC 18 Forces Acting on an Object Newton’s Law uses the forces acting on an object are acting on the object are acting on other objects

Example 4.5 Action-reaction PHYS HCC 19 A man of mass M=75.0 kg and woman of mass m=55.0 kg stand facing each other on an ice rink, both wearing ice skates. The woman pushes the man with a horizontal force of F=85.0 N in the positive x-direction. Assume the ice is frictionless. (a)What is the man’s acceleration? (b) What is the reaction force acting on the woman? (c) Calculate the woman’s acceleration.

Gravitational Force Mutual force of attraction between any two objects Expressed by Newton’s Law of Universal Gravitation: Every particle in the Universe attracts every other particle with a force that is directly proportional to the square of the distance between them 20 PHYS HCC

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 w = m g is a special case of Newton’s Second Law g is the acceleration due to gravity g can also be found from the Law of Universal Gravitation Weight is not an inherent property of an object Mass is an inherent property Weight depends upon location 21 PHYS HCC

Example 4.3 Forces of Distant Worlds PHYS HCC 22 (a) Find the gravitational force exerted by the Sun on a 70.0 kg man located at the Earth’s equator at noon, when the man is closest to the Sun. (b) Calculate the gravitational force of the Sun on the man at midnight, when he is farthest from the Sun. (c) Calculate the difference in the acceleration due to the Sun between noon and midnight. (For values, see Table 7.3, page 223)

Application of Newton’s Laws PHYS HCC 23 A Crate being pulled to the right on a frictionless surface.

Assumptions about crate Objects behave as particles Can ignore rotational motion (for now) Masses of strings or ropes are negligible Interested only in the forces acting on the object Can neglect reaction forces 24 PHYS HCC

25 Assumptions about Ropes Ignore any frictional effects of the rope Ignore the mass of the rope The magnitude of the force exerted along the rope is called the tension The tension is the same at all points in the rope

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

27 Free Body Diagram of Crate The force is the tension acting on the box The tension is the same at all points along the rope are the forces exerted by the earth and the ground

Apply Newton’s second Law to Crate PHYS HCC 28

Solving Newton’s Second Law Problems Read the problem at least once Draw a picture of the system Identify the object of primary interest Indicate forces with arrows Label each force Use labels that bring to mind the physical quantity involved Draw a free body diagram If additional objects are involved, draw separate free body diagrams for each object Choose a convenient coordinate system for each object Apply Newton’s Second Law The x- and y-components should be taken from the vector equation and written separately Solve for the unknown(s) 29 PHYS HCC

Example 4.8: Moving a crate PHYS HCC 30 The combined weight of the crate and dolly is 3.00×10 2 N. If the man pulls on the rope with a constant force of 20.0N, what is the acceleration of the system(crate and dolly), and how far will it move in 2.00s? Assume the system starts from rest and that there are no friction forces opposing the motion?

Group problem: Running car PHYS HCC 31 (a) A car of mass m is on an icy driveway inclined at an angle Ɵ =20.0°, as in Figure. Determine the acceleration of the car, assuming the incline is frictionless. (b) If the length of the drive way is 25.0 m and the car starts from rest at the top, how long does it take to travel to the bottom? (c) What is the car’s speed at the bottom?

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 (since the acceleration is zero) 32 PHYS HCC

Equilibrium cont. Easier to work with the equation in terms of its components: This could be extended to three dimensions A zero net force does not mean the object is not moving, but that it is not accelerating 33 PHYS HCC

Example 4.6 A Traffic light at rest PHYS HCC 34 A traffic light weighting 1.00×10 2 N hangs from a vertical cable tied to two other cables that are fastened to a support as in Figure 4.14a. The upper cables make angles of 37.0° and 53.0° with the horizontal. Find the tension in each of the three cables.

Multiple Objects – Example When you have more than one object, the problem- solving strategy is applied to each object Draw free body diagrams for each object Apply Newton’s Laws to each object Solve the equations 35 PHYS HCC

Example 4.11: Atwood’s Machine PHYS HCC 36 Two objects of mass m 1 and m 2, with m 2 >m 1, are connected by a light, inextensible cord and hung over a frictionless pulley, as in Figure 4.20a. Both cord and pulley have negligible mass. Find the magnitude of the acceleration of the system and the tension in the cord.

Forces of Friction When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion This is due to the interactions between the object and its environment This is resistance is called friction 37 PHYS HCC

38 Static friction acts to keep the object from moving If F increases, so does ƒ s If F decreases, so does ƒ s ƒ s  µ s n Use = sign for impending motion only Static Friction, ƒ s

PHYS HCC 39 Kinetic Friction, ƒ k The force of kinetic friction acts when the object is in motion ƒ k = µ k n Variations of the coefficient with speed will be ignored

Some Coefficients of Friction 40 PHYS HCC

Example 4.12 A Block on a Ramp PHYS HCC 41 Suppose a block with a mass of 2.50 kg is resting on a ramp. If the coefficient of static friction between the block and ramp is 0.350, what maximum angle can the ramp make with the horizontal before the block starts to slip down?

The system approach PHYS HCC 42 The objects are rigidly connected. When two objects are considered a system, external force of one objects becomes internal force of the system.

Example 4.15 Two Blocks and a cord PHYS HCC 43 A block of mass m=5.00 kg rides on top of a second block of mass M=10.0 kg. A person attaches a string to the bottom block and pulls the system horizontally across a frictionless surface, as in Figure 4.26a. Friction between the two blocks keeps the 5.00 kg block from slipping off. If the coefficient of static friction is 0.350, (a) what maximum force can be exerted by the string on the 10.0 kg block without causing the 5.00 kg block to slip? (b) Use the system approach to calculate the acceleration.

Group Problem: Two Blocks PHYS HCC 44 Suppose instead the string is attached to the top block in Example Find the maximum force that can be exerted by the string on the block without causing the top block to slip.