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0612345. Version number. Multiple-choice answer sheets: HB pencil only; ink will not work Fill circle completely No extra marks in answer area Erase well to change an answer. J. P. Student. Centre of Mass. Centre of Mass, Centre of Gravity. Serway 12.1-12.2; and parts of 9.5.

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  1. 0612345 Version number Multiple-choice answer sheets: HB pencil only; ink will not work Fill circle completely No extra marks in answer area Erase well to change an answer J. P. Student Physics 1D03

  2. Centre of Mass • Centre of Mass, Centre of Gravity Serway 12.1-12.2; and parts of 9.5 Physics 1D03

  3. Example: The uniform beam (weight 400 N, length L) is supported by a pin at one end, and a scale that can be placed anywhere along the beam. What will the scale read if it is scale • at the end of the beam (shown)? • at the centre of the beam? • at distance ¼ L from the pin? • at distance ¾ L from the pin? Physics 1D03

  4. Where should we choose the “pivot point” when calculating torques? Answer: In Statics problems, it doesn’t matter. Theorem: If the net force on a body is zero, then the total torque due to all forces will not depend on the choice of “pivot point”. In particular, if Fnet = 0, then if tnet = 0 for one “pivot point”, tnet is also zero for every other pivot point. Physics 1D03

  5. x2 x1 m1 m2 xCM Centre of Mass, Centre of Gravity Two particles on the x axis; total mass, M = m1 + m2 . The centre of mass (CM) is defined as the location: The centre of mass is an “average position” of the mass. Physics 1D03

  6. For many particles: (Recall the position vectorr has components x, y, z in R3.) Symmetry:The center of mass of any symmetric object lies on the axis of symmetry and on any plane of symmetry. For a uniform object, CM = center of geometry and massis distributed evenly around this point Physics 1D03

  7. Examples: The CM is on the symmetry axes. The CM can be outside the object. (It is on the line between the centres of the two rectangles which make up the “L”.) 2L/3 L/3 The CM is closer to the more massive object CM m 2m Physics 1D03

  8. Example: Three particles m1=m2=1.0kg and m3=2kg are located like so: y(m) m3 2 rCM 1 m1 m2 x(m) 3 2 1 What is the center of mass of the system ? Physics 1D03

  9. Quiz:Baseball bat A baseball bat is sawn in half at its centre of mass. Which piece is heavier? • The short piece • The long piece • Both pieces have the same mass. Physics 1D03

  10. For an object, like the baseball bat, torques due to the gravitational forces are zero around the center of mass • But, this is not the point around which mass is evenlydistributed b/c the object has a non-uniform mass distribution • The lighter side of the bat has a larger moment of inertia,since it is longer Physics 1D03

  11. Centre of Gravity The CM is also the location of the centre of gravity. When we consider the rotational equilibrium of a rigid body, we can treat the gravitational force as if it were a single force applied at the centre of mass. A suspended object will hang with its CM vertically below the point of suspension. CM mg Physics 1D03

  12. Quiz:A hemisphere on a ramp. A uniform solid hemisphere is placed on a ramp. Which of the pictures shows how it rests in equilibrium? • The top picture • The middle picture • The bottom picture Physics 1D03

  13. x3 x2 O x1 m2g m1g m3g xCM O Mg Centre of Gravity Why can we place the gravitational force at the CM? Calculate the torque (about O) due to the three weights on the x axis: Now calculate the torque as if a single, total weight were placed at the CM: but The two methods give the same torque ! Physics 1D03

  14. Replacing the real gravitational forces by a single force, equal to the total weight, and placed at the centre of gravity (which may or may not have any mass at this point), will not change the total gravitational torque (about any pivot). This means that the external forces needed to hold a rigid body in equilibrium can be calculated as if gravity were a single force applied at the centre of gravity. In a uniform gravitational field, the centre of gravity is at the centre of mass (eg: a sphere). Physics 1D03

  15. ? Example: how far can a pile of bricks lean without falling over? Is it possible for the top brick to be entirely past the edge of the table? Physics 1D03

  16. Example: how far can a pile of bricks lean without falling over? If the centre of gravity of the entire pile is past the edge of the table, there will be a clockwise torque about the edge of the table (point E), and the whole stack will tip (rotate clockwise) about E. E If the centre of gravity of the pile is not past the edge of the table, the gravitational torque (about E) will be counterclockwise, and the stack will be stable (at least it will not tip at E).The 4th brick can be entirely past the edge Physics 1D03

  17. Summary: Center of mass: Gravitational forces may be replaced by a single force (weight) located at the CM For a uniform object, CM = center of geometry and massis distributed evenly around this point Physics 1D03

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