NEWTON S LAWS OF MOTION, EQUATIONS OF MOTION, & EQUATIONS OF MOTION FOR A SYSTEM OF PARTICLES

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1 NEWTON S LAWS OF MOTION, EQUATIONS OF MOTION, & EQUATIONS OF MOTION FOR A SYSTEM OF PARTICLES Objectives: Students will be able to: 1. Write the equation of motion for an accelerating body. 2. Draw the free-body and kinetic diagrams for an accelerating body. Newton s Laws of Motion Newton s Law of Gravitational Attraction Equation of Motion o For A Particle or System of Particles

2 APPLICATIONS The motion of an object depends on the forces acting on it. A parachutist relies on the atmospheric drag resistance force of her parachute to limit her velocity. Knowing the drag force, how can we determine the acceleration or velocity of the parachutist at any point in time? This has some importance when landing!

3 APPLICATIONS (continued) A freight elevator is lifted using a motor attached to a cable and pulley system as shown. How can we determine the tension force in the cable required to lift the elevator and load at a given acceleration? This is needed to decide what size cable should be used. Is the tension force in the cable greater than the weight Is the tension force in the cable greater than the weight of the elevator and its load?

4 NEWTON S LAWS OF MOTION (Section 13.1) 1) The motion of a particle is governed by Newton s three laws of motion. First Law: A particle originally at rest, or moving in a straight line at constant velocity, will remain in this state if the resultant force acting on the particle is zero. Second Law: If the resultant force on the particle is not zero, the particle experiences an acceleration in the same direction as the resultant force. This acceleration has a magnitude proportional to the resultant force. Thi d L M lf f i d i b Third Law: Mutual forces of action and reaction between two particles are equal, opposite, and collinear.

5 NEWTON S LAWSOFMOTION (continued) The first and third laws were used in developing the concepts of statics. Newton s second law forms the basis of the study of dynamics. Mathematically, Newton s second law of motion can be written F = ma where F is the resultant unbalanced force acting on the particle, and a is the acceleration of the particle. The positive scalar m is called the mass of the particle. Newton s second law cannot be used when the particle s speed approaches the speed of light, or if the size of the particle is extremely small (~ size of an atom).

6 NEWTON S LAW OF GRAVITATIONAL ATTRACTION Any two particles or bodies have a mutually attractive gravitational force acting between them. Newton postulated the law governing this gravitational force as F = G(m 1 m 2 /r 2 ) where F = force of attraction between the two bodies, G = universal constant tof gravitation ti, m 1, m 2 = mass of each body, and r = distance between centers of the two bodies. When near the surface of the earth, the only gravitational force having any sizable magnitude is that between the earth and the body. This force is called the weight of the body.

7 MASS AND WEIGHT It is important to understand the difference between the mass and weight of a body! Mass is an absolute property of a body. It is independent of the gravitational field in which it is measured. The mass provides a measure of the resistance of a body to a change in velocity, as defined by Newton s second law of motion (m = F/a). / ) The weight of a body is not absolute, since it depends on the gravitational field in which it is measured. Weight is defined as W = mg where g is the acceleration due to gravity.

8 UNITS: SI SYSTEM VS. FPS SYSTEM SI system: In the SI system of units, mass is a base unit and weight htis a derived dunit. Typically, mass is specified din kilograms (kg), and weight is calculated from W = mg. If the gravitational ti acceleration (g) is specified in units of m/s 2, then the weight is expressed in newtons (N). On the earth s surface, g can be taken as g = 9.81 m/s 2. W (N) = m (kg) g (m/s 2 ) => N = kg m/s 2 FPS System: In the FPS system of units, weight is a base unit and mass is a derived unit. Weight is typically specified in pounds (lb), and mass is calculated from m = W/g. If g is specified in units of ft/s 2, then the mass is expressed in slugs. On the earth s surface, g is approximately 32.2 ft/s 2. m (slugs) = W (lb)/g g( (ft/s 2 ) => slug = lb s 2 /ft

9 EQUATION OF MOTION (Section 13.2) The motion of a particle is governed by Newton s second law, relating the unbalanced forces on a particle to its acceleration. If more than one force acts on the particle, the equation of motion can be written F = F R =ma where F R is the resultant force, which is a vector summation of all the forces. To illustrate the equation, consider a particle acted on by two forces. First, draw the particle s free-body diagram, showing all forces acting on the particle. Next, draw the kinetic diagram, showing the inertial force ma acting in the same direction as the resultant force F R.

10 INERTIAL FRAME OF REFERENCE This equation of motion is only valid if the acceleration is measured in a Newtonian or inertial frame of reference. What does this mean? For problems concerned with motions at or near the earth s surface, we typically assume our inertial frame to be fixed to the earth. We neglect any acceleration effects from the earth s rotation. For problems involving satellites or rockets, the inertial frame of reference is often fixed to the stars.

11 EQUATION OF MOTION FOR A SYSTEM OF PARTICLES (Section 13.3) The equation of motion can be extended to include systems of particles. This includes the motion of solids, liquids, or gas systems. As in statics, there are internal forces and external forces acting on the system. What is the difference between them? Using the definitions of m = m i as the total mass of all particles and a G as the acceleration of the center of mass G of the particles, then ma G = m i a i. The text shows the details, but for a system of particles: F = ma G where F is the sum of the external forces acting on the entire system.

12 KEY POINTS 1) Newton s second law is a law of nature -- experimentally proven, not the result of an analytical proof. 2) Mass (property of an object) is a measure of the resistance to a change in velocity of the object. 3) Weight (a force) depends on the local gravitational field. Calculating the weight of an object is an application of F=ma a, i.e., ie W=mg g. 4) Unbalanced forces cause the acceleration of objects 4) Unbalanced forces cause the acceleration of objects. This condition is fundamental to all dynamics problems!

13 PROCEDURE FOR THE APPLICATION OF THE EQUATION OF MOTION 1) Select a convenient inertial coordinate system. Rectangular, normal/tangential, or cylindrical coordinates may be used. 2) Draw a free-body diagram showing all external forces applied to the particle. Resolve forces into their appropriate components. 3) Draw the kinetic diagram, showing the particle s inertial force, ma. Resolve this vector into its appropriate components. 4) Apply the equations of motion in their scalar component form and solve these equations for the unknowns. 5) It may be necessary to apply the proper kinematic relations to generate additional equations.

14 EXAMPLE 10 kg Given: A 10-kg block is subjected to the force F=500 N. A spring of stiffness k=500 N/m is mounted against the block. When s=0, the block is at rest and the spring is uncompressed. The contact surface is smooth. Find: Draw the free-body and kinetic diagrams of the block. Plan:

15 EXAMPLE 1 Given: A 10-kg block is subjected to the force F=500 N. A spring of stiffness k=500 N/m is mounted against the block. When s=0, the block is at rest and the spring is uncompressed. The contact surface is smooth. Find: Draw the free-body and kinetic diagrams of the block. Plan: 1) Define an inertial coordinate system. 2) Draw the block s free-body diagram, showing all external forces applied to the block in the proper directions. 3) Draw the block s kinetic diagram, showing the inertial force vector ma in the proper direction.

16 EXAMPLE 1 (continued) Solution: 1) An inertial x-y frame can be defined as fixed to the ground. 2) Draw the free-body diagram of the block: F=500 (N) W = 10 g The weight force (W) acts through the 3 block s center of mass. F is the applied 4 F s =500 s (N) load odand F s =500s (N) is the spring force, y where s is the spring deformation. The x normal force (N) is perpendicular to the N surface. There is no friction force since the contact surface is smooth. 3) Draw the kinetic diagram of the block: 10 a The block will be moved to the right. The acceleration can be directed to the right if the block is speeding up or to the left if it is slowing down.

17 EXAMPLE 2 Given: The block and cylinder have a mass of m. The coefficient of kinetic friction at all surfaces of contact is μ. Block A is moving to the right. Find: Plan: Draw the free-body and kinetic diagrams of each block. 1) Define an inertial coordinate system. 2) Draw the free-body diagrams for each block, showing all external forces. 3) Draw the kinetic diagrams for each block, showing the inertial forces.

18 EXAMPLE 2 Solution: (continued) 1) An inertial x-y frame can be defined as fixed to the ground. 2) Draw the free-body diagram of each block: Block B: y Block A: y 2T W A = mg x F fa = μn A N A T The friction force opposes the motion of block A relative to the surfaces on which it slides. 3) Draw the kinetic diagram of each block: Block B: Block A: ma A ma B x W B = mg

19 EQUATIONS OF MOTION: RECTANGULAR COORDINATES Objectives: Students will be able to: 1. Apply Newton s second law to determine forces and accelerations for particles in rectilinear motion. Equations of Motion Using Rectangular (Cartesian) Coordinates

20 APPLICATIONS If a man is trying to move a 100 lb crate, how large a force F must he exert to start moving the crate? What factors influence how large this force must be to start moving the crate? If the crate starts moving, is there acceleration present? What would you have to know before you could find these answers?

21 APPLICATIONS (continued) Objects that move in air (or other fluid) have a drag force acting on them. This drag force is a function of velocity. If the dragster is traveling with a known velocity and the magnitude of the opposing drag force at any instant is given as a function of velocity, can we determine the time and distance required for dragster to come to a stop if its engine is shut off? How?

22 RECTANGULAR COORDINATES (Section 13.4) The equation of motion, F = m a, is best used when the problem requires finding forces (especially forces perpendicular to the path), accelerations, velocities, or mass. Remember, unbalanced forces cause acceleration! Three scalar equations can be written from this vector equation. The equation of motion, being a vector equation, may be expressed in terms of its three components in the Cartesian (rectangular) coordinate system as F = ma or F x i + F y j + F z k = m(a x i + a y j + a z k) or, as scalar equations, F x = ma x, F y = ma y, and F z = ma z.

23 PROCEDURE FOR ANALYSIS Free Body Diagram (always critical!!) Establish your coordinate system and draw the particle s free body diagram showing only external forces. These external forces usually include the weight, normal forces, friction forces, and applied forces. Show the ma vector (sometimes called the inertial force) on a separate diagram. Make sure any friction forces act opposite to the direction of motion! If the particle is connected to an elastic linear spring, a spring force equal to k s should be included on the FBD.

24 PROCEDURE FOR ANALYSIS (continued) Equations of Motion If the forces can be resolved directly from the free-body diagram (often the case in 2-D problems), use the scalar form of the equation of motion. In more complex cases (usually 3-D), a Cartesian vector is written for every force and a vector analysis is often best. A Cartesian vector formulation of the second law is F = ma or F x i + F y j + F z k = m(a x i +a y j +a z k) Three scalar equations can be written from this vector equation. You may only need two equations if the motion is in 2-D.

25 Kinematics PROCEDURE FOR ANALYSIS (continued) The second law only provides solutions for forces and accelerations. If velocity or position have to be found, kinematics equations are used once the acceleration is found from the equation of motion. Any of the kinematics tools learned in Chapter 12 may be needed to solve a problem. Make sure you use consistent positive coordinate directions as used in the equation of motion part of the problem!

26 a P/c = 1.5 m/s 2 EXAMPLE 3 Given: The 100 kg mine car is hoisted up the incline. The motor M pulls in the cable with an acceleration of 1.5 m/s 2. Find: The acceleration of the mine car and the tension in the cable. Plan: Draw the free-body and kinetic diagrams of the car. Using a dependent motion equation, determine an acceleration relationship between cable and mine car. Apply the equation of motion to determine the cable tension.

27 Solution: EXAMPLE 3 (continued) 1) Draw the free-body and kinetic diagrams of the mine car: y W = mg x 30 N 3T = ma C Since the motion is up the incline, rotate the x-y axes. Motion occurs only in the x-direction. We are also neglecting any friction in the wheel bearings, etc., on the cart.

28 EXAMPLE 3 (continued) 2) The cable equation results in s p + 2 s c = l t Taking the derivative twice yields a p + 2 a c = 0 (eqn. 1) The relative acceleration equation is a p = a c + a p/c As the motor is mounted on the car, a p/c = -1.5 m/s 2 So, a p = a c -1.5 m/s 2 (eqn. 2) a P/c = 1.5m/s 2 P/c s c s p Solving equations 1 and 2, yields a C =0.5 m/s 2

29 EXAMPLE 3 (continued) 3) Apply the equation of motion in the x-direction: a P/c = 4ft/s 2 + F x = ma x => 3T mg(sin30 ) = ma x => 3T (100)(sin 30 ) = (100) (0.5) => T = 33.3 N

30 EXAMPLE 4 Given: W A = 10 N W B = 20 N v oa = 2 m/s μ k = 0.2 Find: v A when A has moved 1 m to the right. Plan: This is not an easy problem, so think carefully about how to approach it!

31 EXAMPLE 4 (continued) Solution: Free-body and dkinetic diagrams of B: 2T = W B m B a B Apply the equation of motion to B: + F y = m a y W B 2T = m B a B 10 2T = 20 a B (1)

32 EXAMPLE 4 (continued) Free-body and kinetic diagrams of A: W A T = m A a A F = μ k N N Apply the equations of motion to A: + F y = m a y = 0 N=W A =10N F = μ k N = 2 N + F x = m a x F T = m A a A 2 T = 10 a A (2)

33 EXAMPLE 4 (continued) Now consider the kinematics. s A x y Datums s B Constraint equation: s A + 2 s B = constant or v A + 2 v B = 0 Therefore a A + 2a B = 0 a A = 2 a B (3) (Notice a A is considered positive to the left and a B is positive downward.)

34 EXAMPLE 4 (continued) Now combine equations (1), (2), and (3). 12 T = = 4N 3 a A = 0.2 m/s 2 Now use the kinematic equation: (v 2 = (v 2 A ) 0A ) +2 a A A( (s A s 0A ) (v A ) 2 = (2) 2 +2 (-0.2)(-1) v A = 2.09 m/s

35 EXAMPLE 13.2 A 10-kg projectile is fired vertically upward from the ground, with an initial velocity of 50 m/s. Determine the maximum height to which it will travel if (a) atmospheric resistance is neglected; and (b) atmospheric resistance is measured as F D = 0.01 v 2 N, where v is the speed of the projectile at any instant, measured in m/s.

36 EQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES Today s Objectives: Students will be able to: 1. Apply the equation of motion using normal and tangential coordinates. Equation of Motion in n-t Coordinates

37 APPLICATIONS Race tracks are often banked in the turns to reduce the frictional forces required to keep the cars from sliding up to the outer rail at high speeds. If the car s maximum velocity and a minimum coefficient of y friction between the tires and track are specified, how can we determine the minimum banking angle (θ) required to prevent the car from sliding up the track?

38 APPLICATIONS (continued) The picture shows a ride at the amusement park. The hydraulically-powered y arms turn at a constant rate, which creates a centrifugal force on the riders. We need to determine the smallest angular velocity of the cars A and B so that the passengers do not loose contact with the seat. What parameters do we need for this calculation?

39 APPLICATIONS (continued) Satellites are held in orbit around the earth by using the earth s gravitational pull as the centripetal force the force acting to change the direction of the satellite s velocity. Knowing the radius of orbit of the satellite, we need to determine the required speed of the satellite to maintain this orbit. What equation governs this situation?

40 NORMAL & TANGENTIAL COORDINATES (Section 13.5) When a particle moves along a curved path, it may be more convenient to write the equation of motion in terms of normal and tangential coordinates. The normal direction (n) always points toward the path s center of curvature. In a circle, the center of curvature is the center of the circle. The tangential direction (t) is tangent to the path, usually set as positive in the direction of motion of the particle.

41 EQUATIONS OF MOTION Since the equation of motion is a vector equation, F =ma ma, it may be written in terms of the n & t coordinates as F t u t + F n u n + F b u b = ma t +ma n Here F t & F n are the sums of the force components acting in the t & n directions, respectively. This vector equation will be satisfied provided d the individual id components on each side of the equation are equal, resulting in the two scalar equations: F t =ma t and F n =ma n. Since there is no motion in the binormal (b) direction, we can also write F b = 0.

42 NORMAL AND TANGENTIAL ACCERLERATIONS The tangential acceleration, a t = dv/dt, represents the time rate of change in the magnitude of the velocity. Depending on the direction of F t, the particle s speed will either be increasing or decreasing. The normal acceleration, a n = v 2 /ρ, represents the time rate of change in the direction of the velocity vector. Remember, a n always acts toward the path s center of curvature. Thus, F n will always be directed toward the center of the path. Recall, if the path of motion is defined d dy as y = f(x), the radius of curvature at dx ρ = any point can be obtained from [1 + ( ) 2 ] 3/2 d 2 y dx 2

43 SOLVING PROBLEMS WITH n-t COORDINATES Use n-t coordinates when a particle is moving along a known, curved path. Establish the n-t coordinate system on the particle. Draw free-body and kinetic diagrams of the particle. The normal acceleration (a n ) always acts inward (the positive n-direction). The tangential acceleration (a t ) may act in either the positive or negative t direction. Apply the equations of motion in scalar form and solve. It may be necessary to employ the kinematic relations: a t = dv/dt = v dv/ds a n = v 2 /ρ

44 EXAMPLE 1 Given:At the instant θ = 45, the boy with a mass of 75 kg, moves a speed of 6 m/s, which is increasing at 0.5 m/s 2. Neglect his size and the mass of the seat and cords. The seat is pin connected to the frame BC. Find: Horizontal and vertical reactions of the seat on the boy. Plan: 1) Since the problem involves a curved path and requires finding the force perpendicular to the path, use n-t coordinates. Draw the boy s free-body and dkinetic diagrams. 2) Apply the equation of motion in the n-t directions.

45 Solution: EXAMPLE 1 (continued) 1) The n-t coordinate system can be established on the boy at angle 45. Approximating the boy and seat together as a particle, the free-body and kinetic i diagrams can be drawn. Free-body diagram t W R x n 45 R y = Kinetic diagram t ma t n ma n

46 EXAMPLE 1 (continued) 2) Apply the equations of motion in the n-t directions. (a) F n = ma n => R x cos 45 R y sin 45 +W sin 45 = ma n Ui Using a n = v 2 /ρ = 6 2 /10, W = 75(9.81) N, and m = 75 kg, we get: R = 2 x cos R y sin = (75)(6 /10) (1) (b) FF t = ma t => R x sin 45 + R y cos 45 W cos 45 = ma t we get: R x sin 45 + R y cos = 75 (0.5) (2) Using equations (1) and (2), solve for R x, R y. R x = 217 N, R y =572 N

47 Example 2 Given: A 800 kg car is traveling over the hill having the shape of a parabola. When it is at point A, itistravelingat9m/sand is traveling at and increasing its speed at 3 m/s 2. Find: The resultant normal force and resultant frictional force exerted on the road at point A. Plan: 1) Treat the car as a particle. Draw the free-body and kinetic diagrams. 2) Apply the equations of motion in the n-t tdirections. 3) Use calculus to determine the slope and radius of curvature of the path at point A.

48 EXAMPLE 2 (continued) Solution: 1) The n-t coordinate system can be established on the car at point ita. Treat tthe car as a particle and draw the free- body and kinetic diagrams: n F N W θ θ t = ma n W = mg = weight of car N = resultant normal force on road F = resultant friction force on road n ma t t

49 EXAMPLE2 (continued) 2) Apply the equations of motion in the n-t directions: F n = ma n => W cos θ N = ma n Using W = mg and a n = v 2 /ρ ρ = (9) 2 /ρρ => (800)(9.81) cos θ N = (800) (81/ρ) => N = 7848 cos θ 64800/ρ (1) F t = ma t =>Wsinθ θ F = ma t Using W = mg and a t = 3 m/s 2 (given) => (800)(9.81) sin θ F = (800) (3) => F = 7848 sin θ 2400 (2)

50 EXAMPLE 2 (continued) 3) Determine ρ by differentiating y = f(x) at x = 80 m: y = 20(1 x 2 /6400) => dy/dx = ( 40) x / 6400 => d 2 y/dx 2 = ( 40) / 6400 [1 + ( dy ) 2 ] 3/2 [1 + ( 0.5) 05) dx 2 ] 3/2 ρ = = d 2 y x = 80 m dx 2 Determine θ from the slope of the curve at A: = m dy θ dx tan θ = dy/dx x = 80 m θ = tan -1 (dy/dx) = tan -1 (-0 0.5) = 26.6

51 GROUP PROBLEM SOLVING (continued) From Eq.(1): N = 7848 cos θ / ρ = 7848 cos (26.6 ) / = 6728 N From Eq.(2): F = 7848 sin θ 2400 = 7848 sin (26.6 )) 2400 = 1114 N

52 Today s Objectives: Students will be able to: 1. Analyze the kinetics of a particle using cylindrical coordinates. EQUATIONS OF MOTION: CYLINDRICAL COORDINATES Equations of Motion Using Cylindrical Coordinates Angle Between Radial and Tangential Directions

53 APPLICATIONS The forces acting on the 100-lb boy can be analyzed using the cylindrical coordinate system. How would you write the equation describing the frictional force on the boy as he slides down this helical slide?

54 APPLICATIONS (continued) When an airplane executes the vertical loop shown above, the centrifugal tif lforce causes the normal lf force (apparent weight) on the pilot to be smaller than her actual weight. How would you calculate the velocity necessary for the pilot to experience weightlessness at A?

55 CYLINDRICAL COORDINATES (Section 13.6) This approach to solving problems has some external similarity to the normal & tangential method just studied. However, the path may be more complex or the problem may have other attributes that make it desirable to use cylindrical coordinates. Equilibrium equations or Equations of Motion in cylindrical coordinates (using r, θ, and z coordinates) may be expressed in scalar form as:... F = = m(r 2 r ma r r θ ).... F θ = ma θ = m (r θ 2 r θ ).. F z =ma z =mz

56 CYLINDRICAL COORDINATES (continued) If the particle is constrained to move only in the r θ plane (i.e., the z coordinate is constant), then only the first two equations are used (as shown below). The coordinate system in such a case becomes a polar coordinate system. In this case, the path is only a function of θ. θ... F = = 2 r ma r m(r rθ )... F θ = ma θ = m(rθ 2rθ. ) Note that a fixed coordinate system is used, not a bodycentered system as used in the n t approach.

57 TANGENTIAL AND NORMAL FORCES If a force P causes the particle to move along a path defined by r = f (θ ), the normal force N exerted by the path on the particle is always perpendicular to the path s tangent. The frictional force F always acts along the tangent in the opposite direction of motion. The directions of N and F can be specified relative to the radial coordinate by using angle ψ.

58 DETERMINATION OF ANGLE ψ The angle ψ, defined as the angle between the extended radial line and the tangent to the curve, can be required to solve some problems. It can be determined from the following relationship. tan ψ = r dθ dr = r dr/dθ If ψ is positive, it is measured counterclockwise from the radial If ψ is positive, it is measured counterclockwise from the radial line to the tangent. If it is negative, it is measured clockwise.

59 EXAMPLE 1 Given: The ball (P) is guided along the vertical circular. path. W.. = 0.5 lb, θ = 0.4 rad/s, θ = rad/s 2, r c = 04f 0.4 ft Find: Plan: Force of the arm OA on the ball when θ = 30.

60 EXAMPLE 1 Given: The ball (P) is guided along the vertical circular. path. W.. = 0.5 lb, θ = 0.4 rad/s, θ = rad/s 2, r c = 04f 0.4 ft Find: Force of the arm OA on the ball when θ = 30. Plan: Draw a FBD. Then develop the kinematic i equations and finally solve the kinetics problem using cylindrical coordinates. Solution: o Notice that r = 2r c cos θ,therefore: eeoe:.. r = -2r c sin θθ..... r = -2r c cos θ θ 2 2r c sin θ θ

61 EXAMPLE 1 (continued) Free Body Diagram and Kinetic Diagram : Establish the r, θ inertial coordinate system and draw the particle s free body diagram. mg = ma θ ma r N s 2θ N OA

62 EXAMPLE 1 (continued) Kinematics: at θ = 30 r = 2(0.4) cos(30 ) = ft. r = - 2(0.4) sin(30 )(0.4) = ft/s.. r = - 2(0.4) cos(30 )(0.4) 2 2(0.4) sin(30 )(0.8) = ft/s 2 Acceleration components are... a r = r rθ 2 = (0.693)(0.4) ) 2 = ft/s a θ = rθ + 2rθ = (0.693)(0.8) + 2(-0.16)(0.4) = ft/s 2

63 EXAMPLE 1 (continued) Equation of motion: r direction F r = ma r 0.5 N s cos(30 ) 0.5 sin(30 ) = (-0.542) N s = lb = ma θ ma r

64 EXAMPLE 1 (continued) Equation of motion: θ direction F θ = ma θ 0.5 N OA sin(30 ) 0.5 cos(30 ) = (0.426) N OA = 0.3 lb ma θ ma r =

65