Chapters 5-8. Dynamics: Applying Newton s Laws

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1 Chaptes 5-8 Dynamics: Applying Newton s Laws Systems of Inteacting Objects The Fee Body Diagam Technique Examples: Masses Inteacting ia Nomal Foces Masses Inteacting ia Tensions in Ropes. Ideal Pulleys 2D Dynamics: Cicula Motion

2 Systems of Inteacting Objects The Fee Body Diagam Technique In geneal objects ae not isolated: they moe as pat of systems of objects In the peious lectue set we e aleady intoduced two contact foces that can intemediate the inteaction between the aious pats of a system: nomal foces and tensions in wies When exeted between pats of a system, these foces ae called intenal foces: they cannot modify the motion of a system as a whole, but they can edistibute the motion between the diffeent pats The Fee Body Diagam technique offes a stategy to analyze systematically the motion of such a system. In geneal we ll follow its steps: 1. Impat the system into fee bodies 2. Build the foce diagam fo each fee body 3. Select a system of coodinate fo each fee body and wite out Newton s 2 nd Law fo each one of them 4. Sole the esulting simultaneous equations fo the unknowns of the poblem: in 2D, a complex of n bodies will esult in 2n independent equations: n along x and n along y. These can be completed with specific definitions. The equations will contain foces, masses, acceleations and constant paametes In ou analysis we ll make as usually simplifying assumptions: fo instance, we ll assume that the fee bodies and opes cannot be defomed, and we ll neglect the mass of opes and pulleys called ideal pulleys

3 Poblem 1. Multiple bodies inteacting by suface contact: Thee cates with masses m 1, m 2 and m 3 sit on a ough suface with coefficient of kinetic fiction µ. The cates ae pushed by a foce F hoizontally fom left. The system acceleates hoizontally. a) Sketch complete fee body diagams fo each cate. b) Based on these diagams, wite Newton s 2nd Law symbolically fo each cate, along some xy-diections. c) Calculate the hoizontal acceleation of the system in tems of gien quantities, using the equations witten in pat b). Analyze the final equation you obtained fo, and suggest an agument that could hae gien you the espectie elationship without the intemediay step of consideing the fee body diagams. F m 1 m 2 m 3

passing aound thee ideal pulleys, as in the figue.

4 Poblem: 2. Multiple bodies inteacting by opes passing aound pulleys: Two boxes with masses m 1 and m 2 ae connected by a massless ope passing aound thee ideal pulleys, as in the figue. When the system is eleased fom est (configuation A), the mass m 1 moes upwad and m 2 downwad. a) Sketch complete fee body diagams fo each mass, and the tensions on the pulleys b) Based on these diagams, wite Newton s 2nd Law symbolically fo each mass, along some xy-diections. c) Calculate the acceleation of the two masses m 1 m 2

5 Dynamics of Cicula Motion Centipetal Foces The centipetal acceleation is caused by a net centipetal foce. By Newton s 2 nd Law 2 F ma m path F a The centipetal foce is the esultant of the adial components of all the foces acting on the object Theefoe, to calculate it, add the adial components of all foces acting on the paticle moing in a cicle, taking the inwad components to be positie and outwad components negatie Then, the centipetal acceleation is gien by Newton s 2 nd Law applied along the adius as aboe Ex: 3 foces acting on a mass m moing in a cicle 2 F F2 F1 m F 3 F 2 Note that, if the foce is pependicula on the adius, it doesn t contibute to the centipetal foce F 1 F 1 F 2

6 Execise: How Angelina Died by Bad Physics In the moie Wanted, bullets ae cued by skilled tattooed assassins. Fo instance, Angelina kills heself by fiing a bullet in a cicle passing though the skulls of some bald dudes befoe hitting he. Say that the bullet has a mass m = 4.2 g and a muzzle speed of 600 m/s. Also, say that the dudes aanged themseles coneniently in a cicle with adius = 5.0 m. a) How big should be the centipetal foce keeping the bullet on the cicula tajectoy? Meditate about the possible oigin of such a foce and how ealistic is such a scenaio b) How fast should Angelina toss the gun to the sweaty guy in the middle fo the scene to make sense?

7 Dynamics of Cicula Motion Unifom and Nonunifom motion If the tangent components of the foces acting on a paticle moing in a cicle cancel each othe out, only the centipetal foce is non-zeo, the motion is unifom cicula (that is, the speed = constant) Othewise, if besides the centipetal foce thee is a net tangent foce component, the motion is nonunifom cicula (that is, the speed is not constant) In a non-unifom cicula motion, by Newton s 2 nd Law, the net tangent fo is esponsible fo the tangent acceleation Hence, the tangent foce is gien by d Ft mat m dt F a F t a t Changing speed Ex: Going back to the example on the peious slide, we see that the mass m pefoms nonunifom cicula motion, since besides centipetal foce it is acted by a net tangent foce esponsible fo a change in speed d Ft F1 t F2 t F3 m dt F 1 F 3 F 1t F 2t F 2

A tension T acting adially inwad pependicula on elocity b) Based on Newton s 2 nd Law, what is this foce in tems of gien quantities? 2 T ma m T Poblem 3.

8 Execise: A ball of mass m is connected by a sting of length and moed into a cicula tajectoy with constant speed. Let s estimate the foce a peson must exet on the sting to make the ball eole in a hoizontal cicle. a) What kind of centipetal foce is exeted on the ball? A tension T acting adially inwad pependicula on elocity b) Based on Newton s 2 nd Law, what is this foce in tems of gien quantities? 2 T ma m T Poblem 3. Tension as a centipetal foce: Now let s assume that the ball fom the execise aboe is swung in a etical cicle, still with a constant speed. a) Detemine the tension in an abitay point of the cicle whee the adius makes an angle θ with espect to the hoizontal b) Use the esult fom pat (a) to detemine the tension in the sting when the sting is hoizontal, on top of the cicle, and at the bottom of the cicle.

9 Poblems: 4. Nomal as a centipetal foce: A small ca with mass m = 1.6 kg moes in a etical cicle inside a hollow metal cylinde that has a adius of = 5.0 m. Its speed deceases unifomly fom 1 = 22 m/s in point A to 2 = 17 m/s in point B. What is the magnitude of the nomal foce exeted on the ca by the walls of the cylinde in a) point A b) point B 5. Conical pendulum: A bob of mass m is suspended fom a fixed point with a massless sting of length L (i.e., it is a pendulum). What tangential speed must the bob hae, so that it moes in a hoizontal cicle with the sting always making an angle θ fom the etical?

Chapter 7-8 Rotational Motion

Chapter 7-8 Rotational Motion Chapte 7-8 Rotational Motion What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics The Toque,