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Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces
CONCEPT: KINETIC FRICTION ● Kinetic friction (____) is a resisting force that occurs when rough surfaces _____________________ against each other . - ? 𝒌 tries to _______ all motion between the surfaces, so its direction is always ____________ of 𝑣 ⃗ . - μ k = _____________ of kinetic friction : measure of roughness between 2 surfaces, unitless # between __ & __, Perfectly smooth surfaces : μ k is [ ZERO | LOW | HIGH ] Ice rubbing on ice : μ k is [ ZERO | LOW | HIGH ] Cinderblock on cinderblock : μ k is [ ZERO | LOW | HIGH ] EXAMPLE : A 10-kg box moves on a flat surface at 2 m/s . The coefficient of kinetic friction between the box and the surface is 0.4 . Calculate (a) the kinetic friction force acting on the box and (b) the acceleration of the box . FORCES 1) Draw FBD 2) Write 𝜮𝑭 = 𝒎𝒂 3) Solve 𝒗 ⃗⃗ ? 𝒌 = _______ 𝒗 ⃗⃗ Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 1
PROBLEM : Pushing a 10-kg toolbox across the floor, you find that the box moves at a constant speed when you push horizontally with a force of 39 N . What is the coefficient of kinetic friction between the floor and the toolbox? A) 0.2 B) 0.4 C) 2.5 PROBLEM : You push on a 3-kg box to give it an initial speed of 5 m/s across a floor . If μ k = 0.3, how far does the box travel before coming to a stop? A) 8.6 m B) 2.9 m C) 7.7 m D) 4.3 m FORCES 1) Draw FBD 2) Write 𝜮𝑭 = 𝒎𝒂 3) Solve FORCES 1) Draw FBD 2) Write 𝜮𝑭 = 𝒎𝒂 3) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 2
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EXAMPLE : A 20-kg box moving along the floor has a downward force of 30N acting on it . How hard must you push the box horizontally to keep the box moving at a constant 2m/s if the coefficient of kinetic friction μ k is 0.3? A) 67.8 N B) 58.8 N C) 49.8 N D) 30 N ● Never assume that N = mg , therefore ? 𝒌 = 𝝁(𝒎?) ! Remember to always calculate N using Σ F = ma. FORCES 1) Draw FBD 2) Write 𝜮𝑭 = 𝒎𝒂 3) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 3
CONCEPT: STATIC FRICTION ● Static Friction is another type of friction similar to kinetic friction . EXAMPLE : A 5.1kg block is at rest on the floor . The coefficients of static & kinetic friction are 0.6 and 0.3, respectively . Determine the magnitude of the friction force on the block when you push it with a force of : ● μ s N is a threshold : the force you must overcome to get an object moving, so μ s N is the _____ value of 𝒇 𝒔 : 𝒇 𝒔,______ = _____ - This is NOT always the actual friction acting on an object . To determine if 𝒇 𝒔 vs . 𝒇 𝒌 , compare F to f s,max : IS F STRONG ENOUGH TO GET OBJECT MOVING? NO ( F ___ 𝒇 𝒔,𝒎𝒂𝒙 ) YES ( F ___ 𝒇 𝒔,𝒎𝒂𝒙 ) Object… [ STAYS AT REST | STARTS MOVING ] [ STAYS AT REST | STARTS MOVING ] Friction is… [ STATIC ( 𝒇 𝒔 ) | KINETIC ( 𝒇 𝒌 ) ] [ STATIC ( 𝒇 𝒔 ) | KINETIC ( 𝒇 𝒌 ) ] KINETIC FRICTION STATIC FRICTION 𝒗 ⃗ 𝒇 𝒌 = 𝝁 𝒌 𝑵 𝒇 𝒔,______ = ______ 𝒇 𝒌 𝑭 ● When 𝑣 ___ 0 ● Tries to _________ an object from starting to move ● Direction : __________ to where the object would move without friction . ● When 𝑣 ≠ 0 ● Tries to stop objects already moving ● Direction : opposite of motion ( 𝒗 ⃗ ) 𝑭 a) F = 20N b) F = 40N 𝑭 * 𝑭 =10 𝑭 =20 𝑭 =30 threshold 𝒇 𝒔,𝒎𝒂𝒙 𝑭 =40 𝑭 =50 𝒇 𝒔 = ___ 𝒇 𝒌 = _____ , μ s = coeff. of static friction ; μ s always __ μ k Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 4
PROBLEM : A 5.1 kg block is at rest on the floor . The coefficients of static & kinetic friction are μ s = 0.7 and μ k = 0.5 . Calculate the force needed to get the block moving , and the force needed to keep it moving at constant speed . A) F = 0.014 N; F = 0.01 N B) F = 3.57 N; F = 2.55 N C) F = 35; F = 25 D) Impossible to tell ● Remember : μ s ≥ μ k ! It’s always harder to [ GET | KEEP ] something moving than it is to [ GET | KEEP ] it moving . PROBLEM : A 15 kg block is initially at rest on a horizontal surface . The coefficient of static friction between the block and the surface is μ s =0.7 . How hard must you push down on the block to keep a 300 N horizontal force from moving it? A) 147 N B) 210 N C) 282 N D) 429 N FRICTION 1) Draw FBD 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write 𝜮𝑭 = 𝒎𝒂 4) Solve FRICTION 1) Draw FBD 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write 𝜮𝑭 = 𝒎𝒂 4) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 5
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PROBLEM : A 36N force is needed to start a 7.0 kg box moving across the floor . If the 36.0 N force continues, the box accelerates at 0.70 m/s 2 . What are the coefficients of static and kinetic friction? E) μ s =0.52 and μ k =0.64 F) μ s =0.64 and μ k =0.64 G) μ s =0.52 and μ k =0.45 H) μ s =0.45 and μ k =0.32 FRICTION 1) Draw FBD 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write 𝜮𝑭 = 𝒎𝒂 4) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 6
CONCEPT: CONNECTED SYSTEMS OF OBJECTS WITH FRICTION ● If 2+ objects are connected and friction is NOT negligible, you’ll have to consider the friction on _______ object . - Remember: Connected objects have the same 𝒗 ⃗ and 𝒂 ⃗ ! EXAMPLE : A 10kg block is tied via a string to a 5kg block on a rough table where μ s = 0.5 and μ k = 0.3 . If you pull on the 10kg block with 90N, and the objects start moving, a) Draw FBDs for both blocks ; b) find the acceleration of the blocks . CONNECTED OBJECTS + FRICTION 1) Draw FBD for all obj’s, choose direction of + 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write Σ F=ma, start with simplest (fewest F s) 4) Solve a (EQ Addition / Substitution) 5) Plug a into eq ’s, solve other targets if needed 5 10 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 7
PROBLEM : Two blocks are connected by a cord over a pulley . Block A rests on a rough tabletop . Block B has mass m B =2kg and hangs over the edge of the table . The coefficients of friction between Block A and the tabletop are μ s =0.6 and μ k =0.4 . What is the minimum mass Block A can have to keep the system from starting to move? A) 3.33 kg B) 5 kg C) 32.7 kg CONNECTED OBJECTS + FRICTION 1) Draw FBD for all obj’s , choose direction of + 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write Σ F=ma, start with simplest (fewest F s) 4) Solve a (EQ Addition / Substitution) 5) Plug a into eq ’s, solve other targets if needed Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 8
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[ EXAMPLE ] Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 9
CONCEPT: SOLVING INCLINED PLANE PROBLEMS WITH FRICTION ● You’ll need to solve problems with objects on inclined planes WITH friction . - Remember: To determine if objects begin moving, compare all non-friction forces ALONG the axis of motion to ____ . EXAMPLE : You release a 10kg block on ramp inclined at 37° . The coefficients of friction are μ s = 0.6 and μ k = 0.4 . a) Calculate the friction force acting on the block when it is released . b) Calculate the block’s acceleration . INC. PLANES + FRICTION 1) Draw FBD (tilt x&y axes on inclines) 2) Determine if 𝒇 = 𝒇 ? or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write 𝜮𝑭 = 𝒎𝒂 4) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 10
PROBLEM : You attempt to push a 20-kg box up a ramp into a moving truck, applying a 110-N force parallel to the incline . The ramp is angled at 15° . The coefficients of friction between the box and the ramp are μ s =0.3 and μ k =0.2 . What are the magnitude and direction of the box's acceleration? A) 0 m/s 2 (no direction) B) 0.1 m/s 2 up the ramp C) 1.1 m/s 2 up the ramp D) 4.5 m/s 2 down the ramp ● If the direction of 𝒇 isn’t known, find the net of all non-friction forces along axis of motion . 𝒇 will be ___________ to that . INC. PLANES + FRICTION 1) Draw FBD (tilt x&y axes on inclines) 2) Determine if 𝒇 = 𝒇 ? or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write 𝜮𝑭 = 𝒎𝒂 4) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 11
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PROBLEM : A 2.0-kg block is launched up 40° ramp at 10 m/s . If the block comes to a stop when it reaches a point 3 m vertically above the bottom of the ramp, calculate the coefficient of kinetic friction between the block and the ramp . A) 0.45 B) 0.51 C) 0.59 D) 0.67 INC. PLANES + FRICTION 1) Draw FBD (tilt x&y axes on inclines) 2) Determine if 𝒇 = 𝒇 ? or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write 𝜮𝑭 = 𝒎𝒂 4) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 12
PROBLEM : You push a 30kg mini-fridge up a 20° incline . Your push is 120N angled 30° above the axis of the incline . If the coefficients of friction between the box and the ramp are μ s =0.3 and μ k =0.2, what is the box's acceleration? A) 0 m/s 2 (no direction) B) 0.1 m/s 2 up the ramp C) 1.1 m/s 2 up the ramp D) 4.5 m/s 2 down the ramp INC. PLANES + FRICTION 1) Draw FBD (tilt x&y axes on inclines) 2) Determine if 𝒇 = 𝒇 ? or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write 𝜮𝑭 = 𝒎𝒂 4) Solve Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 13
CONCEPT: CRITICAL ANGLES ON ROUGH INCLINED PLANES ● For objects on rough inclines, there are 2 special angles called _________ angles : - For both critical angles, 𝒂 = ___ . EXAMPLE : You place a 6kg block on an adjustable ramp, then tilt the angle of the ramp very slowly until the block suddenly starts sliding. If μ s = 0.75, calculate this special angle . ● For critical angles, (1) μ s & θ crit,s and (2) μ k & θ crit,k only depend on each other, not on any other variable (e.g. mass) . 𝜽 𝒄?𝒊?,? = _______ EXAMPLE : Using the same ramp, once the block begins sliding you tilt the ramp so that the block slides down at constant speed. If μ k = 0.31, calculate this special angle . 𝝁 ? = _________ ⇔ 𝒗 ⃗ 2) θ crit,k → block “ slides at constant speed ” : ____ = ____ 𝜽 𝒄?𝒊?,𝒌 = _______ ⇔ 𝝁 𝒌 = _________ 1) θ crit,s → block “ STARTS sliding ” : ____ = ______ Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 14
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PROBLEM : A 3-kg block is at rest on an adjustable ramp . When the ramp is tilted to a 20° angle, the block slides with a constant velocity . What is the coefficient of kinetic friction between the ramp and the block? A) μ k =0.24 B) μ k =0.36 C) μ k =0.52 D) Not enough information given PROBLEM : In the afternoon, a car is parked on a street that runs down a steep hill, at an angle of 35.0° relative to the horizontal . After a snowstorm hits the area, and the road becomes icy and just slippery enough for the car to slide downhill . What is the coefficient of static friction between the car tires and the icy road? A) μs=0.47 B) μs=0.70 C) μs=1.54 D) Not enough information given 𝜽 𝒄?𝒊?,? = tan −1 (𝜇 ? ) 𝝁 ? = tan (𝜃 𝑐?𝑖?,? ) 𝜽 𝒄?𝒊?,𝒌 = tan −1 (𝜇 𝑘 ) 𝝁 𝒌 = tan (𝜃 𝑐?𝑖?,𝑘 ) 𝜽 𝒄?𝒊?,? = tan −1 (𝜇 ? ) 𝝁 ? = tan (𝜃 𝑐?𝑖?,? ) 𝜽 𝒄?𝒊?,𝒌 = tan −1 (𝜇 𝑘 ) 𝝁 𝒌 = tan (𝜃 𝑐?𝑖?,𝑘 ) Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 15
CONCEPT: CONNECTED OBJECTS ON INCLINED PLANES WITH FRICTION ● Some problems will combine multiple objects on ramps with friction ! Usually, you’ll know which kind of friction is acting . - To solve, use all the problem-solving steps for systems of objects, inclined planes, and friction . EXAMPLE : Two blocks are connected by a cable & massless pulley . Block B pulls Block A, which moves up the 30° incline . Block B weighs 100N (m B = 10.2kg), block A weighs 40N (m A = 4.1kg) . If μ k = 0.15, find the acceleration of the system . CONNECTED OBJECTS + INCLINED PLANES + FRICTION 1) Draw FBD for all obj’s, choose direction of + 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write Σ F=ma, start with simplest (fewest F s) 4) Solve a (EQ Addition / Substitution) 5) Plug a into eq ’s, solve other targets if needed B Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 16
PROBLEM : Two blocks made of different materials, connected by a string, slide down a 30° inclined plane . Block A has mass 8kg, and the coefficient of static friction between Block A and the incline is 0.35 . Block B has mass 4kg, and the coefficient of friction between block B and the plane is 0.25 . After the blocks are released, find the tension in the cord . A) 1.23 N B) 2.21 N C) 1.67 N D) 2.28 N CONNECTED OBJECTS + INCLINED PLANES + FRICTION 1) Draw FBD for all obj’s, choose direction of + 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write Σ F=ma, start with simplest (fewest F s) 4) Solve a (EQ Addition / Substitution) 5) Plug a into eq ’s, solve other targets if needed θ Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 17
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PROBLEM : Two blocks, A and B, sit on back-to-back rough inclined planes and are connected to each other by a cable . The angles of the planes are θ A =15° and θ B =30° . The masses of the blocks are m A =2kg and m B =5kg, and μ k =0.2 . When the blocks are released from rest and begin moving, what is the magnitude of their acceleration? A) 2.47 m/s 2 B) 1.02 m/s 2 C) 3.45 m/s 2 D) 10.5 m/s 2 CONNECTED OBJECTS + INCLINED PLANES + FRICTION 1) Draw FBD for all obj’s, choose direction of + 2) Determine if 𝒇 = 𝒇 𝒔 or 𝒇 𝒌 from text or : If Σ Fs on axis of motion > f s,max, 𝒇 = 𝒇 𝒌 3) Write Σ F=ma, start with simplest (fewest F s) 4) Solve a (EQ Addition / Substitution) 5) Plug a into eq ’s, solve other targets if needed θ A θ B Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 18
CONCEPT: INTRO TO UNIFORM CIRCULAR MOTION ● In Uniform Circular Motion (UCM), objects move in a circular path with __________________ . - 𝒗 ⃗ changes direction in UCM ; 𝒗 ⃗ at any point is called the ____________ velocity ( 𝒗 ⃗ 𝑻 ) . - 𝒂 ⃗ (“centripetal” = center -seeking) points towards _________ of the path ( 𝒂 ⃗ 𝑪 or 𝒂 ⃗ 𝒓𝒂? ) . - 𝑹 is the distance from the edge of the path to the center, or the ___________ of the path . EXAMPLE : You move at constant 5 m/s when you turn into a circle of radius 10m . Calculate your centripetal acceleration . 𝒂 ? = Units : [ _____ ] Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 19
PROBLEM : A ball travels on a frictionless circular track at 3m/s . The ball cannot have an acceleration greater than 1.5m/s 2 or it will go off the track . What is the smallest radius the circular track can have so that the ball stays on the track? PROBLEM : The Moon travels in a circular orbit of radius R = 3.85×10 8 m around the Earth because of gravity . Because of the large distance, the centripetal acceleration of the Moon is only 0.0026m/s 2 . How fast would the Moon be moving if it suddenly broke free of Earth’s gravity and stopped orbiting? Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 𝟐 𝑹 Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 𝟐 𝑹 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 20
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CONCEPT: CIRCUMFERENCE, PERIOD AND FREQUENCY IN UNIFORM CIRCULAR MOTION ● When objects complete a full ROTATION a.k.a. REVOLUTION a.k.a. CYCLE: - the distance traveled is called the CIRCUMFERENCE ___ = _____ - Period (__) → # of __________ per __________ ; Unit : [ __ ] or [ ] - Frequency (__) → # of __________ per __________ ; Unit : [ ___ = ] or [ ] EXAMPLE : Calculate the period and frequency of your motion if you complete : PROBLEM : Modern windmills usually spin at a rate of 20 Revolutions Per Minute ( R.P.M ) . At this rate, how long does it take for a windmill blade to complete a full rotation? ● Whenever you’re given Revs per Minute (RPMs), to get frequency → 𝒇 = 𝐑𝐏𝐌 𝟔? Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? T = a) 4 rotations in 2 seconds b) 0.5 rotations in 3 seconds 𝒇 = ⇔ Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 21
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CONCEPT: MORE EQUATIONS FOR VELOCITY AND ACCELERATION IN UNIFORM CIRCULAR MOTION ● We can combine the circumference, period, and frequency into equations for 𝒗 𝑻 and 𝒂 𝑪 : EXAMPLE : A ball moves in a circle of radius 10m . Calculate : 𝒗 𝑻 = = OR _________ a) its speed if it takes 60 seconds to complete 100 rotations 𝒂 𝑪 = OR ___________ 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 ⇒ 𝒗 𝑻 = distance time ⇒ 𝑹 b) its centripetal acceleration if it completes 1 rotation every 3 minutes Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 22
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PROBLEM : A 3kg rock spins horizontally at the end of a 2m string at 90 RPM . Calculate its centripetal acceleration . PROBLEM : A big problem for astronauts in space is the lack of gravity ! One way to simulate gravity is to build a space ship with spinning rings attached to it . If a cylindrical space station of diameter = 500m is spun about its axis, how fast in revolutions per minute (RPM) must it turn so the astronauts inside feel an acceleration equal to that of Earth ( g) ? Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 Circ. Motion 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 23
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CONCEPT: INTRO TO CENTRIPETAL FORCES ● LINEAR force problems vs . CIRCULAR / CENTRIPETAL forces are solved the same way! But there are differences : EXAMPLE : A small 3kg block is tied to the end of 2m string and slides around in a circle on a frictionless table . If the block completes a rotation every 4 seconds, calculate the tension on the string . CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = 𝒎𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 +? +? −? −? LINEAR CENTRIPETAL ● Forces along X & Y axes - Solve with 𝛴𝐹 ? = 𝑚𝑎 ? , 𝛴𝐹 ? = 𝑚𝑎 ? ● Forces along ______________ direction, i.e. toward/away from center - Solve with ___________ 𝒗 𝑻 𝒂 𝑪 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 24
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PROBLEM : A small 4kg block is tied to the end of 3m string and slides around in a circle on a frictionless table . Suppose the string will break if the tension exceeds 50N . Find the maximum speed the block can have without breaking the string . A) 37.5m/s B) 6.1m/s C) 8.7m/s D) 2.8m/s PROBLEM : A 0.50-kg ball is tied to the end of a 4m light cord that spins the ball around in a horizontal plane at constant speed with the cord making a 30° angle with the vertical . Determine the ball’s speed . CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = 𝒎𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = 𝒎𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 θ Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 25
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CONCEPT: VERTICAL CENTRIPETAL FORCES ● For Circ. Mo. problems in the vertical plane, mg pulls obj’s downwards, so speed is [ CONSTANT | NOT CONSTANT ] EXAMPLE : You’re sitting in a rollercoaster as it goes in a vertical loop of radius 10m . Your mass is 70kg . At the bottom of the loop, your speed is 30m/s . At the top of the loop, your speed is 20m/s . Calculate the centripetal acceleration and the normal force from the seat on you when you are at the a) bottom of the loop and b) top of the loop . ● There are also “rules” to determine signs of Centripetal Forces : - Forces pointing towards center are [ + | – | 0 ] - Forces pointing away from center are [ + | – | 0 ] - Forces pointing perpendicular ( 90° ) to the direction to center are [ + | – | 0 ] CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = 𝒎𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 26
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PROBLEM : Suppose a 1,800-kg car passes over a bump in a roadway that follows the arc of a circle of radius 20m . What force does the road exert on the car as the car moves over the top of the bump if the car moves at a constant 9 m/s? A) 10350N B) 24930N C) 17640N D) 16830N CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = 𝒎𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹𝒇 ? 𝑻 = ? 𝒇 ⇔ 𝒇 = ? 𝑻 = 𝐑𝐏𝐌 𝟔? 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹𝒇 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 27
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CONCEPT: FLAT CURVE ● When objects travel around horizontal (flat) curves, the force that keeps them in circular motion is _________________ . EXAMPLE : You drive an 800kg car around a flat curve of radius 50m . The coefficient of static friction between the car and the road is 0.5 . a) Write an expression for the maximum speed you can drive the car around the curve without slipping on the road . b) Calculate this maximum speed . CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = 𝒎𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹? ? 𝑻 = ? ? ⇔ ? = ? 𝑻 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹? 𝑹 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 28
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PROBLEM : A truck can go around a flat curve of radius 150m with a maximum speed of 32m/s before slipping . Calculate the maximum speed it can go around a tighter curve of radius 75m . A) 32 m/s B) 515 m/s C) 3.8 m/s D) 22.7 m/s CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = 𝒎𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹? ? 𝑻 = ? ? ⇔ ? = ? 𝑻 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹? Flat Curve: 𝒗 ? = ?𝑹𝝁 𝒔 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 29
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CONCEPT: BANKED CURVE ● Unlike flat curve problems, Banked Curve problems involve objects travelling in circular motion on frictionless inclines . - Because the centripetal direction is horizontal , we use an ____________ coordinate system (regular X & Y axes) . - WITHOUT friction, objects accelerate centripetally from the ____________ Force (x-component) . EXAMPLE : An 800kg racecar on a racetrack drives around a banked, frictionless curve inclined 37° above the horizontal . The radius of the curve is 200m . Find the exact speed of the car such that it moves without sliding up OR down the incline . 𝝁 𝜽 EQ Flat Curve Banked Curve CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = ?𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹? ? 𝑻 = ? ? ⇔ ? = ? 𝑻 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹? Flat Curve: 𝒗 ? = ?𝑹𝝁 ? Banked Curve: 𝒗 ? = ?𝑹?𝒂?𝜽 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 30
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PROBLEM : A bobsled turn banked at 78° is taken at 24 m/s . Assume it is ideally banked and there is no friction between the ice and the bobsled . Calculate the centripetal acceleration of the bobsled . A) 1100 m/s 2 B) 2.08 m/s 2 C) 46.1 m/s 2 D) 1.92 m/s 2 CENTRIPETAL FORCES 1) Draw FBD 2) Write 𝜮𝑭 𝒄 = ?𝒂 𝒄 (rewrite 𝒂 𝑪 ⇒ 𝒗 ? / 𝑹 ) 3) Solve Circ. Motion / Centripetal Forces 𝒂 𝑪 = 𝒗 𝑻 ? 𝑹 = 𝟒𝝅 ? 𝑹 𝑻 ? = 𝟒𝝅 ? 𝑹? ? 𝑻 = ? ? ⇔ ? = ? 𝑻 𝒗 𝑻 = 𝑪 𝑻 = ?𝝅𝑹 𝑻 = ?𝝅𝑹? Flat Curve: 𝒗 ? = ?𝑹𝝁 ? Banked Curve: 𝒗 ? = ?𝑹?𝒂?𝜽 Giancoli Calc - 4th edition - Physics for Scientists and Engineers Ch 05: Using Newton's Laws: Friction, Circular Motion, Drag Forces Page 31
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