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Torque:Center of Mass (Unit 5).pdf

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Department
Physics
Course
PHYSICS 1L03
Professor
Reza Nejat
Semester
Fall

Description
LUCY Human evolution and Physics Bipedalism, Quadrupedalism Who is Lucy? Concepts: Center of Mass Torque Force Equilibrium Chapters: 4, 5 Pages: 95-101, 122-141 PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Lucy Figure 04.12c • A partial hominid skeleton, discovered at Hadar, Ethiopia, Africa in 1974. •Aliu. •.atlt,d •Crytr bipedal locomotion •irprl humans (~3.2 million years) PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Lucy Lucy •W dwhik “Lucy” was bipedal •4%omlltn (walking upright)? of “Lucy” indicates that she was bipedal. •W rureieefr effective and efficient bipedal posture –WasShe? and locomotion? • Lucy’s entire skeleton was •L’mpewtnCiaze reconstructed from the (quadruped, knuckle walking), and remains that were Human (biped, upright walking) discovered. PHYS 1L03, Fall 2013, McMaster University, R.NejaPHYS 1L03, Fall 2013, McMaster University, R.Nejat Quadrupedalism vs. Bipedalism Quadrupedalism vs. Bipedalism sInets ela •uail are considered: •Dsir ght need considerable change to the shape of the pelvis bones? –Doesthischangeinpelvisshapecreatesa change in the position of the center of mass. •Dsihote function of gluteal (abductor) muscles? PHYS 1L03, Fall 2013, McMaster University, R.NejaPHYS 1L03, Fall 2013, McMaster University, R.Nejat Quadrupedalism vs. Bipedalism Center of Mass •oseuteow sltetn equivalent the physics of: single particle: Center of Mass –Riojilecallandsofortharesystems •Wits?a of particles –Whatisweightline? •Tremains constant in time. •Wit? –Themassofanobjectcanbedistributed –Whatdoesforcedo? uniformly or non-uniformly •Aladtitemiustr)pe –Howcanwequantifythemassdistribution? •Wit? –Whatdoestorquedo? •e center of mass (CM) of a system is the point at which all the mass of the system may be •Witiillu considered to be concentrated. PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass •Tstm may be either: se –agroupofobjects(particles) extended object like our body. –oranextendedobjectlikeourbody. •olit of the center of mass of the system is the same as if the mass of the system is concentrated at that point. PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass •Fxodt, •Tcfsfa different parts move with different speed. a ball thrown in the air •Smtadiid follows a parabolic terms of the position and motion of the center of mass of the body (point particle Model). path •W sti easily because the ball is a simple, point- like object PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass •Tcflih )alreafohTt•n also follows a a straight-line path, as though it were a particle, but the wrench is rotating around its parabolic path when the hammer is thrown in the air, but this is harder to observe center of mass hemolhrosphhtoT PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass •Mytjtc http://hippie.nu/~unicorn/tut/xhtml/ http://hippie.nu/~unicorn/tut/xhtml/ PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass: Math Example 1: Center of mass •hotf center of mass of a •wo.,s system of two particles of unequal masses positions 6.0 m and – 4.0 m respectively from is: the origin as shown below. Where is the center of mass of these two masses? -4 -2 0 2 4 6 x ▯m1x1▯ 2 2 m 1 ▯1m x 2 2 CM m1▯ m2 xCM ▯ m 1 m 2 •sotir the system. PHYS 1L03, Fall 2013, McMaster University, R.Nejat PPHHYYSS11LL0033,,FFaallll22001132,,MMccMMaasstteerrUUnniivveerrssiittyy,,RR..NNeejjaatt Center of Mass: Math Center of Mass qtfiIs masses, the x coordinate of the center of mass y is: m 2 1y2 m1 mx ▯ 1x 2 x1▯ x 2 xi x CM ▯ ▯ 2 x1 m ▯ m 2 i m1 1 ▯ m 2 2▯ m 3 3▯ ...▯ m n n m i xCM ▯ •S, the position of the center of mass in this m 1 ▯ m 2 ▯ m3 ▯ ... m n case is merely the average position of the YCM ▯ m1y 1▯ m1y 2▯ m 3 3 ▯..▯ m n n system. m 1 m ▯2m ▯ 3 ..▯ mn PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Example 1: Center of mass Center of Mass • Three masses, 4.0 kg, 6.0 kg and 3.0 kg are •Tefsify placed at positions 6.0 m, – 2.0 m and 4.0 m position in space described mathematically respectively from the origin as shown below. as: Where is the center of mass of these masses? x ▯ m 1 1m x2▯2 x 3 3 ... ▯ ▯ m ii CM m ▯m ▯m ▯ ... m m1 m2 1 2 3 ▯ i (11, y x2y2 -4 -2 CM 0 2 4 6 m y m3 m 1 1 m 2 ▯2m x 3 3 yCM ▯ m1y1▯ m2y 2▯m 3 3▯... ▯ ▯ ii (3 3 y ) xCM ▯ m 1▯m 2▯m 3▯... ▯ mi m1▯ m 2m 3 •baedijtie many small particles, having specific mass and specific coordinates. PPHHYYSS11LL0033,,FFaallll22001132,,MMccMMaasstteerrUUnniivveerrssiittyy,,PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass Center of Mass ,umttro •ymbjemis having irregular shape, the center of mass may be distributed uniformly (i.e. constant density), evaluated by suspending as shown below: the center of mass is at the geometric center of the object msth body of any shape can be located by suspending it from at least two points. PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Example 1: Center of mass of the Arm Center of mass of the Arm elnFin etiri The masses of the upper arm, the lower arm and ▯ mii ▯ miyi the hand are 2.5 kg, 1.6 kg and 0.64 kg respectively. CM ▯ ▯ mi , yCM▯ ▯ mi Their center of mass are at 0.18 m above the elbow, and 0.12 m and 0.40 m to the right of elbow respectively, as illustrated below. ▯ ▯ m x ▯xCM ▯ i i ▯ ▯ mi 0.18 m ▯ elbow ▯y ▯ ▯ miyi 0.12 m ▯ CM ▯ mi 0.40 m PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Center of Mass (review) Physics: Torque •heCs)a of a system is the point at which all the mass of the system may •Qeis: be considered to be concentrated. •hltf the center of mass of the system is the same as if the mass of What is Torque? the system is concentrated at that point. What does Torque do? •nljte center, nor will it divide an object into equal mass points. PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Pop-Quiz : Torque Physics: Torque Q: Where on a door do you push to open it? A C E B D A: Far from the hinge. Ywtte this door most easily (i.e. with the smallest possible force), along which line should you push? PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Physics: Torque Physics: Torque •Truittdnyfct •Tqe, ▯ (Greek tau), due to a force F applied at a distance r from the axis of produce rotation about an axis. rotation (pivot, hinge, fulcrum), at an angle ▯ to the radial line is: •Trum yeo nsidered as the F rotational equivalent of force. ▯ ▯ r Fsin▯ F sin ▯ Axis ▯ F sin▯ ▯ r •Tqeruetileitn , r like Force that produces translational F ▯ Fsin▯ ▯ acceleration •:i•t•) PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Physics: Torque Physics: Torque •qeitifi and point of application of force) x •Teu ▯ ,sign convection for direction: (component of force perpendicular to this line) = (r) x (Fsin▯) F F sin ▯ • ▯ > 0, if rotation is counter clockwise. Axis ▯ • ▯ < 0, if rotation is clockwise. r PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Physics: Torque (Review) Physics: Torque • The ability of a force to cause a rotation or a ▯ We quantify the effectiveness of a force twisting motion depends on: in creating rotation using the quantity – The magnitude, F, of the force. called torque. – The distance, r, from the pivot to the point at which the force is applied. – The angle, ▯,atwhc ihthefoiapild ▯ Torque plays the same role in rotational motion that force plays in ▯ = r F sin▯ F F translational motion. ▯ Axis Axis ▯ ▯ = r F sin▯ r r Larger Torque Smaller Torque PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Physics: Calculating Torque ▯ Physics: Calculating Torque ▯ F F ▯ F ▯ = F r sin▯ r ▯ = r F ▯ ▯ Axis Axis ▯ r = r sin▯ Le F ▯ F sin▯ F▯ ▯ ▯ ver e = r ar l r sn m , ce ti F ▯ F cos▯ ▯ = F r ▯ ▯ For of ac F is the perpendicular component of the force F ▯ The distance r ▯ is called Lever arm which contributes to the torque. (moment arm). F ▯s the parallel component of the force F which passes through the axis and does not contribute to The Lever arm is the perpendicular the torque. distance to the line of action of the force. PPHHYYSS11LL0033,,FFaallll22001132,,MMccMMaasstteerrUUnniivveerrssiittyy,PHYS 1L03, Fall 2013, McMaster University, R.Nejat Example 4: Torque Torque: Conceptual Question stdgir•sis •emhitf leg at an angle of 30.0 from the horizontal with an string, which makes angle ▯ with the vertical. 89 N weight attached to the ankle. The distance Which of the following are true? between the ankle weight and the hip joint is 84 cm. What is the torque due to the ankle weight on the The gravitational torque on the pendulum about the pivot point: leg. pivot a. depends on the length of R b. Is zero at the max displacement ▯ R of the swing c. Is constant since g is constant m d. Is smaller when the angle is smaller Try to work through it at home. A way of tesging! PHYS 1L03, Fall 2013, McMaster University, R.Nejat PHYS 1L03, Fall 2013, McMaster University, R.Nejat Physics: Mechanical equilibrium Physics: Mechanical equilibrium •Aojdiib mechanical •arn extended object to be in static or equilibrium if it has neither translational dynamic mechanical equilibrium, the acceleration nor rotational acceleration following two conditions must be satisfied: –itcanstillbetranslatingand/orrotating –The net external force must be zero: at constant speeds (dynamic ▯ mechanical equilibrium) ▯ F ▯ 0 –Thenetexternaltorquemustbezero: •Aoniit static mechanical equilibrium if it is neither rotating
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