Ying Yi PhD Chapter 4 Forces and Newton’s Laws of Motion 1 PHYS HCC

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

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

M versus m PHYS HCC 11

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 13 PHYS HCC

Example 4.1 Pushing a Stalled Car PHYS HCC 14 Two people are pushing a stalled car, as Figure 4.5a indicates. The mass of the car is 1850 kg. One person applies a force of 275 N to the car, while the other applies a force of 395 N. Both forces act in the same direction. A third force of 560 N also acts on the car, but in a direction opposite to that in which the people are pushing. This force arises because of friction and the extent to which the pavement opposes the motion of the tires. Find the acceleration of the car.

Example 4.2 Applying Newton’s 2 nd Law PHYS HCC 15 A man is stranded on a raft (mass of man and raft=1300 kg), as shown in Figure 4.6a. By paddling, he causes an average force of 17 N to be applied to the raft in a direction due east (the +x direction). The wind also exerts a force on the raft. This force has a magnitude of 15 N and points 67° north of east. Ignoring any resistance from the water, find the x and y components of the raft’s acceleration.

Group Problem: Horses Pulling a Barge PHYS HCC 16 Two horses are pulling a barge with mass 2.00×10 3 Kg along a canal, as shown in Figure. 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 17 PHYS HCC

18 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 19 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 20 More Action-Reaction pairs is the force the Earth exerts on the object is the force the object exerts on the earth

PHYS HCC 21 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 4.4 Action and reaction PHYS HCC 22 Suppose that the mass of the spacecraft in figure 4.7 is m S =11000 kg and that the mass of the astronaut is m A =92 kg. In addition, assume that the astronaut pushes with a force of =+36 N on the spacecraft. Find the accelerations of the spacecraft and the astronaut.

Group Problem: Third Law PHYS HCC 23 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 24 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 25 PHYS HCC

Group Problem: Forces of Distant Worlds PHYS HCC 26 (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 next page)

Useful planetary data PHYS HCC 27

Application of Newton’s Laws PHYS HCC 28 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 29 PHYS HCC

30 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 31 PHYS HCC

32 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 33

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) 34 PHYS HCC

Example : Moving a crate PHYS HCC 35 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 36 (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) 37 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 38 PHYS HCC

Example 4.12 Replacing an Engine PHYS HCC 39 An automobile engine has a weight, whose magnitude is W=3150 N. this engine is being positioned above an engine compartment, as Figure 4.28a illustrates. To position the engine, a worker is using a rope. Find the tension in the supporting cable and the tension in the positioning rope.

Group Problem: A Traffic light at rest PHYS HCC 40 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.

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 41 PHYS HCC

42 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 43 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 44 PHYS HCC

Example: Static Friction PHYS HCC 45 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?

Group Problem: Kinetic Friction PHYS HCC 46 A sled and its rider are moving at a speed of 4.0 m/s along a horizontal stretch of snow. The snow exerts a kinetic frictional force on the runners of the sled, so the sled slows down and eventually comes to a stop. The coefficient of kinetic friction is What is the displacement x of the sled?

Homework PHYS HCC 47 5,9,13,17,25,29,35,43,45,48,59

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 48 PHYS HCC

Example 4.18 Accelerating blocks PHYS HCC 49 Block 1 (mass m 1 =8.00 kg) is moving on a frictionless 30.0° incline. This block is connected to block 2 (mass m 2 =22.0 kg) by a massless cord that passes over a massless and frictionless pulley. Find the acceleration of each block and the tension in the cord.

Group Problem: Atwood’s Machine PHYS HCC 50 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.