Understanding Tension: Concepts, Problems, and Solutions

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University of British Columbia**We aren't endorsed by this school

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SCIENCE DSCI100

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Law

Date

Dec 10, 2024

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In this lecture (L09), we will coverTensionSlinky drop: conceptual questionsHooke’s lawPreview: What sort of problems can we solve with what we know now? (wewill work on some of those in the next two lectures)Problem solving ideas and strategies

Tension, the chain modelThe statement “the magnitude of this action-reaction pair is the tension” might beeasier to understand if you consider a chain instead of a rope.The force between any two consecutive links is the tension at that point.

More precisely: The magnitude of the two action-reaction forces between any twoconsecutive links is the tension.Action-reaction force pairs arealways equal. In addition, if thelinks aremassless, the forces oneach must balance exactly. Thismeans that all forces shown areequal in magnitude!Tension is constant along amassless chain (or rope).A massless, non-stretching chain/rope is a magic device for transmitting a forcefrom one place to another along its length.With pulleys, we can also change itsdirection!

Last class we asked about the tension inthe rope in this situation:-!Possible answers were:(A)Tension is zero because they pullwith equal and opposite forces(B)Tension must 200N because the twoforces add to the total tension(C)Tension must be 100N, but I can’texplainLet’s add:(D)Replacing the rope with the chain, we see that the force pulling right on theright-most link is 100N, and this 100N is transmitted, link by link, all the way tothe left, to the other student. Tension is 100N.

Clicker Question 37WQ5.10:What is the tension in the rope measured by the spring scale in each of the two cases? Explain your answer!A)T1 = 98N, T2 = 98NB)T1 = 49N, T2 = 49NC)T1 = 98N, T2 = 49ND)T1 = 49N, T2 = 98N

Three more things about tension:Tension is not a force; it is an internal strain on the rope. The forces occur at theends of the rope, causing the tension (same for compression on a rod or a spring).Imagining cutting the rope. Tension is equal to the magnitude of the force you haveto supply at the newly cut end in order to keep things from sliding away.Corollary: wherever the rope ends, there must be a force acting on it whose magni-tude is equal to the tension and which is acting along the direction of the rope andaway from it.And an important general force “fact”:Forces such as pushing and pulling only act on the object that is actually beingtouched.

Interlude: slinky drop (something fun for a change)Slinky DropWhat happens to the bottom end of a suspended slinky immediately after you let go at the top?A) it dropsB) it remains motionlessC) it risesD) something elseWe will come back to this question (and vote) in a moment.

Clicker Question 38Consider a hanging metal slinky, as shown on the last slide. Why are the coils atthe top separated further apart than those at the bottom?(A)bottom half pulls on the top half more than top half pulls on the bottom half(B)slinky has more weight near the top(C)tension is higher near the top(D)tension is the same everywhere, but gravity pulls the slinky apart more at thetop(E)bottom half of the slinky has less weight than the top half

Clicker Question 39Slinky DropWhat happens to the bottom end of a suspended slinky immediately after you let go at the top?A) it dropsB) it remains motionlessC) it risesD) something elseAnswer in the video: https://www.youtube.com/watch?v=uiyMuHuCFo4

The slinky was our first object that was not rigid. Rigid objects don’t deform underinternal strains (caused by external forces): they don’t bend, stretch, shear, rip,crack, etc. . .The simplest example of a non-rigid body is a massless spring in 1d.Massless (or light) implies thatspring does not compress, stretch or sag under its own weight, only underexternal forces andspring does not have an inertia, so the net force on it must be zero and thetension (or compression) is constant along its lengthNote: The slinky was not massless, so it’s much more complicated.Note 2: The rigid object is an idealization: all objects deform under forces, but weignore it most of the time.

A massless spring follows Hooke’s law: when under compressionFspring, it stretchesby an amountΔx, withFspring=-kΔxFspring>0: compression, withΔx <0Fspring<0: tension, withΔx >0Spring ForceAnswer:Whenacteduponbyanexternalcompressingforcemg=(1.5kg)(9.8m/s2),the spring’s length changesΔx=xcompressed-xuncompressed= (0.35-0.6)m =-0.25mk=-mg/Δx(Note: see how (-) cancels.)

Types of forces we understandGravity (~Fg=m~g)Normal forces (~N, prevent objects from occupying the same space; A acts onB in the direction normal the interface between A and B, and away from A;arbitrarily large (for rigid bodies); go to zero when objects separate)Static friction (stops objects from sliding; maximum magnitudeFS,max=μsN)Kinetic friction (due to objects sliding; direction opposes motion;Fk=μkN)Tension-related forces (for thin, massless ropes and springs, tension is constantalong the length; force at the ends points in the direction of the rope/spring;magnitude of the tension is arbitrarily large for non-stretchy rope/chain; for aspring (or stretchy rope)Fspring=-kΔx).We will add one more this week: air dragWith these, we can ask you to solve a largevariety of problems.

WQ5.12)Consider the Atwood machine, which consists of a rope running over a pulley, with two objects of different mass attached. Assume that we can neglect the mass of the pulley and the mass of the rope. We also assume that the pulley has no friction.If m1= 3 m2, what is the acceleration expressed as a fraction of g?CQ5.12a) Which of the following free-body diagrams best represents the forces experienced by the two blocks?T1m1*gm2*gT2A)T1m1*gm2*gT2B)T1m1*gm2*gT2C)Bonus example: Two blocks are on a surface with negligible friction. The blocks are connected by a string (neglect its mass). If the M = 300 g and the tension in the rope is 0.75 N, what is the magnitude of the force F? Polling question: The two tension forces are not equal, but the system (traffic light) is in equilibrium. Which equations must be satisfied so that the net force is zero?1) |!1| + |!2| − "= 0 2) !1,!+ !2,!− "= 0 3) !1,!− !2,!= 04) !1,"+ !2,"− "= 0 5) |!1|− |!2|= 0A.Equation 1 onlyB. Equation 5 onlyC. Equations 3 and 4D.Equations 1 and 5E. Equations 2 and 4Example:What is the tension in the slackline?Step 1. Draw a free-body diagram for a person is standing on a slackline.Step 2. Calculate the tension in the rope. Assume that the angles between the rope and the horizontal direction are 20° on both sides and that the mass of the person is 65 kg.WQ6.8b)What is the acceleration in case b? Use g=10m/s2.A)3.0 m/s2B)4.3 m/s2C)7.0 m/s2D)10.0 m/s2E)13.0 m/s2WQ6.12: What is the maximum speed (in km/h) at which the car could theoretically go over the top of the hill without losing contact with the road? Assume that the radius of curvature is 30 m.WQ6.10: Ball on a string 1. In this classic demo, a ball is forced to undergo circular motion by the tension in a string. The ball’s motion is in the horizontal plane. Which of the following free-body diagrams is most correct for the ball?vA)WTB)WTC)WTND)WTNCQ6.1: A car is rounding a corner on a banked curve at constant speed. There is ice in the curve (no friction), but the car is not sliding up or down. With respect to the rear view shown, in what direction is the net force on the car? A.Horizontally pointing to the center of the curve.B.Parallel to the slope downhill.C.Perpendicular (normal) to the slope.D.In between horizontal and parallel to the slope.WQ6.13: When an airplane changes direction, its wings are tilted as shown. The lift force “L” acting on an airplane’s wings act only in a direction perpendicular to the wings.a)Draw the free-body diagram for the plane at the position shown below (flying into the paper). Think about: what is keeping the plane at the same altitude (height above ground); what makes the plane move in a circle?b)An airplane is flying in a horizontal circle with an airspeed of 140 m/s. The wings of the plane are tilted at 400with respect to the horizontal (x-) direction. What is the radius of the circle?

CQ5.1: A person tries to pull a heavy box with tension T, but the box does not move. What is the magnitude of the static friction force?A)μsm gB)μkm gC)TD)T – m aE)otherCQ5.2: You place a block on a bookshelf. Then you start to lift the shelf on one side, as shown below. You slowly tilt the shelf (i.e. increase the angle), until the block just starts to slide. Once the block starts to slide, you keep the angle constant. After the block begins to slide, its acceleration is A)zero (it slides with constant velocity)B)non-zero (and it slides faster and faster)C)non-zero (and it slides slower and slower)CQ5.3: A block of mass mis sliding on a table with an initial speed of v0. Due to friction, the block comes to a stop after 0.50 m. What’s the stopping distance for two identical blocks, attached to each other, that are sliding with the same initial speed on the same table? Assume the same μkin both situations.A)0.25 mB)0.50 mC)1.0 mD) otherWQ6.15:Identical blocks are being pulled by either one or two forces of magnitude F, but are not moving. Rank the static friction force in each situation from largest to smallest. A)F > E > D > C > A > BB)D = F > A = E > C > BC)C > A = D = E = F > BD)D = F > A = E > B = CWQ6.16:Compare two situations: in situation 1(left) an object slides down a rough (there is friction) inclined plane with some non-zero acceleration a1. In situation 2 (right), the same object slides up the same incline. As the mass moves up the incline, its acceleration is a2. Compare the acceleration in the two cases and explain!A) a1 = a2B) a1 < a2C) a1 > a2D) a2 must be pointing upif v is pointing up the slopeWQ6.19a) A mass m is pulled along a rough table at constant velocitywith a non-zero external force Fext. The magnitudes of the forces on the free-body diagram have not been drawn with correct relative lengths, but the directions are correct. Explain the correct relationships between the forces:a) In the horizontal directionA) Fext, x > FfricB)Fext, x< FfricC) Fext, x= FfricWQ6.19b) A mass m is pulled along a rough table at constant velocitywith a non-zero external force Fext. The magnitudes of the forces on the free-body diagram have not been drawn with correct relative lengths, but the directions are correct. Explain the correct relationships between the forces:b) In the vertical directionA) F!"#,$= 0 B) F!"#,$= NC) F!"#,$= m g - N D) F!"#,$= m g + NE) F!"#,$= m gWQ6.25: Consider a skydiver jumping out of the plane. In mid-air, there are two forces acting on the skydiver: force of gravity FGand drag force D.a) Explainhow these two forces lead to a maximum speed. b)Use the free-body diagram on the right (the forces have equal magnitude) to derive an equation for the maximum speed (also called ‘terminal velocity’). Q6.26: A physics textbook claims that a mouse would not get hurt if it fell off the roof of a high skyscraper. Calculate its maximum falling speed. Remember that its maximum speed occurs when the mouse is no longer accelerating. As physicists, we tend to simplify things, so we will model our mouse as a small rectangle of dimensions 6.0 cm x 3.5 cm and mass 17 g. Assume that the drag coefficient for the furry mouse is CD= 0.9. Use g = 9.80 m/s2.

The three steps to solving Dynamics problems1. Draw a free body force diagram.You usually need one for every object. You can also draw them for any collectionof objects together as one system. Ignore forces internal to the object/collection.2.Apply Newton’s laws and any formulas you have for the forces (eg,Fk=μkN,Fg=mg, etc. . .).Do this for every object or collection of objects, and for every direction. You candefine your coordinates to match the geometry (for example, point the x-axis alonga slanted ramp instead of horizontaly).3. Math it up.You might need to use what you learned about kinematics earlier in the course.

I’m overwhelmed by this problem solving!What can I rely on?MathLogicLaws of physics, such as Newton’s lawsFacts and equations derived in classif the assumptions made in thederivation are satisfiedPay close attention to definitions and assumptions.In a theorem, the assumptions are as important as the conclusion.Avoid trying to solve problems by analogy with other problems. It’s really easy toget wrong answers this way.

Clicker Question 40WQ5.13 (modified): Consider cases (a) and (b). The two blocks are on very smooth surfaces with negligible friction and are being pulled by tension forces. If block A has a weight of 20N, how do the tensions in the rope compare for case a vs case b?A)Ta > TbB)Ta = TbC)Ta < Tb

Clicker Question 41In which two scenarios is the acceleration of red block the same (ignore friction)?(A)A and D(B)A and B(C)B and D(D)A and C(E)C and D

Consider: Explain how the concept of inertia applies in each caseInertia (related to mass m) = resistance to change in motionc)using a quick pull to tear off d) A crash-test dummy sliding ononly the first grocery bag a car seat.

In lecture (L10/11), we will coverTension with acceleration (clicker question at the end of L09)A bunch of problems (chosen by you)Air dragI will not be solving the problems “all the way”—I will leave the final alge-bra to you. There are two reasons for that:Not going over relatively straight forward algebra leaves us more time to dis-cuss tricky physics.When you finish the question yourself and get the right answer, you have someassurance that you understood the solution presented. If you can’t finish it,you probably need to ask someone (piazza, office hours, me after class) toexplain something about the solution again.

Clicker Question 41In which two scenarios is the acceleration of red block the same (ignore friction)?(A)A and D(B)A and B(C)B and D(D)A and C(E)C and D

The most requested question isWQ6.13: When an airplane changes direction, its wings are tilted as shown. The lift force “L” acting on an airplane’s wings act only in a direction perpendicular to the wings.a)Draw the free-body diagram for the plane at the position shown below (flying into the paper). Think about: what is keeping the plane at the same altitude (height above ground); what makes the plane move in a circle?b)An airplane is flying in a horizontal circle with an airspeed of 140 m/s. The wings of the plane are tilted at 400with respect to the horizontal (x-) direction. What is the radius of the circle?

The second most requested question (section 101) isWQ6.12: What is the maximum speed (in km/h) at which the car could theoretically go over the top of the hill without losing contact with the road? Assume that the radius of curvature is 30 m.

The second most requested question isWQ6.20:A person holds a 10-kg-block against a vertical wall with force F = 100 N at an angle of 60 degreeswith respect to the horizontal direction. The coefficient of static friction is μs= 0.5. a)What is the friction force on the block (magnitude and direction)?b)What is the minimum force the person needs to apply to hold the block in place?

Similar (conceptualy) to the plane question, and also highly demanded is 6.23. Hereis a warm-up clicker question:CQ6.1: A car is rounding a corner on a banked curve at constant speed. There is ice in the curve (no friction), but the car is not sliding up or down. With respect to the rear view shown, in what direction is the net force on the car? A.Horizontally pointing to the center of the curve.B.Parallel to the slope downhill.C.Perpendicular (normal) to the slope.D.In between horizontal and parallel to the slope.

Q6.233 points possible (graded)A car rounds a curve on a flat road at constant speed. The car and driver have a combined massof and the curve has a radius of . The coefficient of static friction is .Draw a free-body diagramthat shows all forces actingon the car.What is the maximumspeed v with which the carcan round the curve withoutskidding? Use .Enter your answer in .

Continuing 6.23 from last slide:If three passengers get in the car and bring the total mass to , what is the maximumspeed with which the car can round the curve without sliding?Enter your answer in .What is the maximum speed with which the carcan round a ‘bankedʼcurve (the road has a slope asshown in the picture)? Assume that the radius ofthe curve is and the car has a total mass of. The road is icy and there is no friction.The angle is .Enter your answer in .

Another highly requested question (and one that will get me to talk about air drag) isQ6.26: A physics textbook claims that a mouse would not get hurt if it fell off the roof of a high skyscraper. Calculate its maximum falling speed. Remember that its maximum speed occurs when the mouse is no longer accelerating. As physicists, we tend to simplify things, so we will model our mouse as a small rectangle of dimensions 6.0 cm x 3.5 cm and mass 17 g. Assume that the drag coefficient for the furry mouse is CD= 0.9. Use g = 9.80 m/s2. Let’s draw a free body diagram before looking up a formula for air drag.

Air Drag: Air molecules are displaced by object..Data:⇢air= 1.21 kg/m3

Polling question: The two tension forces are not equal, but the system (traffic light) is in equilibrium. Which equations must be satisfied so that the net force is zero?1) |!1| + |!2| − "= 0 2) !1,!+ !2,!− "= 0 3) !1,!− !2,!= 04) !1,"+ !2,"− "= 0 5) |!1|− |!2|= 0A.Equation 1 onlyB. Equation 5 onlyC. Equations 3 and 4D.Equations 1 and 5E. Equations 2 and 4