Nicholas J. Giordano Forces and Motion in Two and Three Dimensions

Newton’s Laws and Motion We will extend Newton’s Laws to multiple dimensions as the foundation for explaining motion We can extend ideas of motion to two- and three-dimensional cases Still interested in displacement, velocity and acceleration Must allow for the vector addition of many quantities Force, acceleration, velocity, and displacement Express directions in terms of a chosen coordinate system Apply the same principles and problem-solving techniques as used with one-dimensional motion Introduction

Using Newton’s Second Law Newton’s Second Law in vector form states Generally start by determining all the individual forces acting on the object Construct a free body diagram Add the individual forces as vectors Use Newton’s Second Law to find the acceleration Once the acceleration is found, it can be used to determine velocity and displacement Section 4.1

Statics Deals with objects at rest We will look at the conditions for translational equilibrium Equilibrium examples use the same approach as the one- dimensional problems previously studied Friction may need to be included in equilibrium problems Section 4.1

Statics and Equilibrium Statics is an area of mechanics dealing with problems in which both the velocity and acceleration are zero The object is also said to be in translational equilibrium Often the “translational” is dropped If the acceleration is zero, then This is the condition for translational equilibrium Section 4.1

Equilibrium Example: Refrigerator Four forces act on the refrigerator: Gravity and normal force in y-direction Force exerted by person (push) and static friction in x-direction Draw a free body diagram (b) and include a coordinate system Section 4.1

Refrigerator Example, cont. Express the forces in terms of their x- and y-components In this case, the forces are aligned along the axes Apply the condition for equilibrium: Σ F x = 0 and Σ F y = 0 For y-direction: N - m g = 0 For x-direction: F push - F friction = 0 Section 4.1

Equilibrium Example: Sled All the forces do not all align with the x- or y-axes Find the x- and y-components of all forces that are not on an axis Applying Newton’s Second Law: Σ F x = T x – F friction = T cos θ – F friction = 0 Σ F y = N – mg + T y = N – mg + T sin θ = 0

Equilibrium Example: Tightrope Walker Both sections of the rope exert a tension force at the center where the walker is standing The walker and the rope are at rest The forces acting at the center are shown in the free body diagram Tension forces on the right and on the left Weight of the walker Section 4.1

Tightrope Walker Example, cont. Choose the usual x-y coordinate axes along the horizontal and vertical directions Express all the forces in terms of the x- and y-components The tensions in both sides of the rope are equal Solve for the unknown quantities Section 4.1

Problem Solving Strategy for Statics Problems Recognize the principle For static equilibrium, the sum of the forces must be zero Use Sketch the problem Show the given information in the picture Include a coordinate system Identify the relationships Use all the forces to construct a free body diagram Express all the forces on the object in terms of their x- and y-components Apply Σ F x = 0 and Σ F y = 0 May also include Σ F z = 0 Section 4.1

Problem Solving Strategy for Statics Problems, cont. Solve Solve all the equations The number of equations must equal the number of unknown quantities Check Consider what your answer means Check that your answer makes sense Section 4.1

Inclines (Hills) The normal force (N) acts perpendicular to the incline (plane) The friction force acts up the incline The motion would tend to be down the incline Friction opposes the motion The force due to gravity acts straight down These forces compose your free body diagram Section 4.1

Inclines, cont. Choose a coordinate system Choose axes parallel and perpendicular to the incline Less components If acceleration is present, the acceleration would be along the incline Find the components of the gravitational force The rest of the forces are along the axes The normal force is not equal to mg The value of N depends on the angle of the hill Section 4.1

Angle of Incline To Not Slip Analysis of the problem indicates the minimum frictional force to keep the object from slipping is F friction = m g sin θ Since this is static friction, F friction ≤ μ static N Assuming it is just in equilibrium (so F friction = μ static N), the angle of the incline at which the object is on the verge of slipping is tan θ = μ s Section 4.1

Equilibrium Example: Flag Determine the tension and angle Two unknowns, so need to look at two dimensions Draw the free body diagram Choose horizontal and vertical directions for your coordinate system Tension has x- and y- components Section 4.1

Flag Example, cont. Write the equations for equilibrium in the x- and y- directions Solve for the unknown quantities Check to be sure the answers make sense Section 4.1

Equilibrium in Three Dimensions Many situations can be simplified by choosing the x-y plane to match the geometry of the problem If three dimensions are necessary, the same basic approach is used Include a corresponding relation for the z-direction: ΣF z = 0 Section 4.1

Projectile Motion Consider objects in motion and the forces acting on them Projectile motion is one example of this type of motion We will ignore the force from air drag For now Components of gravity are F grav, x = 0, F grav, y = - m g Section 4.2

Projectile Motion, cont. Accelerations are a x = 0 and a y = - g The acceleration in both x- and y-directions is constant The motions in the x- and y-directions are independent of each other The motion in the x-direction is constant acceleration with a = 0 The motion in the y-direction is constant acceleration with a = - g This is free fall The relationships among displacement, velocity, acceleration, and time for constant acceleration apply directly to projectile motion Section 4.2

Projectile Motion Example: Rolling off a Cliff The car rolls off the cliff Its initial velocity is directed along the horizontal Choose the coordinate system to be the horizontal and vertical directions Apply the relations for motion with constant acceleration Section 4.2

Independence of x- and y- motions The time it takes for the object to reach the ground is independent of its motion along the x- direction Here, the two balls strike the ground at the same time Section 4.2

Projectile Motion Example: Shooting at a Target If the rifle is fired horizontally, the bullet misses the target The bullet falls a distance Δ y while traveling between the rifle and the target To compensate, the rifle is aimed above the target The value of Δ y depends on the distance to the target and the speed of the bullet Section 4.2

Projectile Motion Example: Baseball Throw The baseball has an initial position of x = 0 and y = h h is the height of the ball when it leaves the bat The ball is hit with an initial velocity of v o at an angle of θ above the horizontal v ox = v o cos θ v oy = v o sin θ Section 4.2

Baseball Example: Trajectory To find the trajectory of the ball, you need the x- and y- components of its motion x = v o (cos θ ) t y = h + v o (sin θ ) t – ½ g t 2 The graphs show x and y as functions of time Section 4.2

Baseball Example: Velocity To describe the motion, the velocity components as functions of time are also needed v x = v o cos θ v y = v o sin θ – g t The plots show the velocities as functions of time Note that v x is constant v y varies linearly with time The slope is -g Section 4.2

Baseball Example: Velocity, cont. The graph shows the components of the velocity at various points along the trajectory The total velocity at any point is the vector sum of its components: Section 4.2

Projectile Motion, Final Notes For a symmetrical trajectory: t lands = 2 t to top Speed at landing is equal to its initial speed The range of the projectile is the horizontal distance it travels Applies only if air drag is negligible Applies only if the motion is symmetrical Maximum range will occur at θ = 45° Section 4.2

Reference Frames A reference frame is an observer’s choice of coordinate system for making measurements It will include an origin Newton’s Laws give a correct description of the motion in any reference frame that moves with a constant velocity Section 4.3

Relative Velocity The two cars are traveling at constant velocities along the x-direction For an observer on the sidewalk, the cars have velocities

Relative Velocity, cont. Consider a reference frame at rest with respect to car 1 The reference frame is defined by x’ and y’ in the previous figure The velocity of car 2 with respect to car 1 is The velocity of car 2 relative to the observer = velocity of car 2 relative to car 1 + velocity of car 1 with respect to the observer This gives the general way velocities in different reference frames are related The same ideas apply to motion in two dimensions Apply the ideas to each component Section 4.3

Further Applications of Newton’s Laws There are many forces that can play a role in Newton’s Laws Since we are dealing with two-dimensional problems, directions can also be found In terms of angles Examples will show how to apply the general problem- solving strategy to specific instances Section 4.4

Newton’s Second Law – General Problem Solving Strategy Recognize the principle Consider all the forces acting on the object Compute the total force If in equilibrium, the total force must be zero If the object is not in equilibrium, set the total force equal to ma Sketch the problem Define a coordinate system to include in your sketch It also needs to contain all the forces in the problem Include all the given information Section 4.4

Newton’s Second Law – General Problem Solving Strategy, cont. Identify the relationships Construct the free body diagram Express all the forces in their components along x and y Apply Newton’s Second Law in component form ΣFx = m ax and ΣFy = m ay If the acceleration is constant along the x- or y-axis or both, apply the kinematic equations Solve Solve the equations for all the unknown quantities The number of equations must equal the number of unknown quantities Section 4.4

Newton’s Second Law – General Problem Solving Strategy, final Check Consider what your answer means Be sure the answer makes sense Section 4.4

Traveling Down a Hill Assume a frictionless surface Draw the free body diagram The force of gravity has x- and y-components Looking at the x-direction, along the hill, gives a x = g sin θ Section 4.4a

Traveling Down a Hill with Friction The forces acting on the sled are The normal force exerted by the road The force of gravity The force of friction between the sled and the hill Use kinetic friction since the sled is slipping relative to the hill Section 4.4

Newton’s Second Law Example: Pulleys and Cables Assume a massless cable Therefore the tension is the same in all points of the cable Choose the x-direction parallel to the string The “positive” direction will be opposite for the two crates Apply Newton’s Second Law to each crate Solve the equations Section 4.4

Newton’s Second Law Example: Cables and Pulleys 2 Apply general problem- solving strategy Draw free body diagrams of each object The x-direction follows the string Friction could be included See example 4.12 Section 4.4

Accelerometer An accelerometer is a device that measures acceleration A rock hanging from a string can act as an accelerometer When the airplane moves with a constant velocity, the string hangs vertically The two forces add to zero T = m g Section 4.5

Accelerometer, cont. When the airplane accelerates, the string will hang at an angle θ Now T cos θ = m g and a x = g tan θ By measuring the angle, you can determine the acceleration Section 4.5

Ear as an Accelerometer When your head accelerates, the gelatinous layer lags behind a small amount Very similar to the rock on the string This lag causes the hair cells to deflect The hair cells send signals to the brain The brain interprets the acceleration Section 4.5

Inertial Reference Frames Analyze the motion of the accelerometer in the plane from inside the plane If the acceleration is zero, the string hangs straight down Agrees with the observation of the observer outside the plane If the acceleration is nonzero to an observer outside the plane, he does not agree with the observer inside the plane The inside observer would observe the angle of the accelerometer, but would calculate an acceleration of zero Newton’s Second Law can only be applied in inertial reference frames Nonaccelerating frames Section 4.5

Projectile Motion with Air Drag With air drag, the maximum range no longer occurs at 45° For a baseball, the maximum range occurs when the ball is projected at an angle of approximately 35° Also depends on the initial speed of the ball Artillery guns are usually aimed much higher than 45° They travel high enough to reach altitudes where the air is not as dense and so drag is reduced A bicycle coasting down a hill reaches a terminal velocity that depends on the angle of the hill and the density of the air Section 4.6

Air Drag and a Bicycle Assume a frictionless bicycle Forces acting on the bicycle along the incline are gravity and air drag The terminal (coasting) velocity of the bicycle will depend on The mass The angle of the hill (incline) The density of the air The frontal area of the cyclist Section 4.6

Newton’s Second Law in Three Dimensions The examples have shown applications of Newton’s Second Law in two dimensions: ∑F x = m a x and ∑F y = m a y This allows you look at forces and accelerations in both directions To deal with a problem that involves three dimensions, add a third equation to the two above ∑F z = m a z The basic approach is the same as in two dimensions