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EXAMPLE: POSITION OF SECOND KID ON SEESAW EXAMPLE: A 4 m-long seesaw 50 kg in mass and of uniform mass distribution is pivoted on a fulcrum at its middle, as shown. Two kids sit on opposite sides of the seesaw. The kid on the left (30 kg) sits on the very edge of the seesaw. How far from the fulcrum should the kid on the right (40 kg) sit, if they want to balance the fulcrum? m 1 m 2? Page 2
PRACTICE: BALANCING A BAR WITH A MASS PRACTICE: A 20 kg, 5 m-long bar of uniform mass distribution is attached to the ceiling by a light string, as shown. Because the string is off-center (2 m from the right edge), the bar does not hang horizontally. To fix this, you place a small object on the right edge of the bar. What mass should this object have, to cause the bar to balance horizontally? m Page 3
PRACTICE: POSITION OF FULCRUM ON SEESAW PRACTICE: Two kids (m,left = 50 kg, m,right = 40 kg) sit on the very ends of a 5 m-long, 30 kg seesaw. How far from the left end of the seesaw should the fulcrum be placed so the system is at equilibrium? (Remember the weight of the seesaw!) m 1 m 2? Page 4
EXAMPLE: MULTIPLE OBJECTS HANGING EXAMPLE: The system of objects shown is in linear and rotational equilibrium, held by light, vertical ropes and light, horizontal rods. Calculate the: (a) tension on all 5 vertical ropes; (b) 2 missing masses (ma and mc). Use g = 10 m/s 2. 4 m 1 m A 1 m 2 m 4 kg C Page 5
EQUILIBRIUM WITH MULTIPLE SUPPORTS When an object in Equilibrium has MULTIPLE supports, we can think of each support point as a potential. m - Therefore, we can write for ANY point support, which means treating it as the. - In fact, we can write for ANY point, even points that are not the or points! - Since you can choose your reference axis in writing equations, you ll want to pick the easier ones. - Remember that forces acting ON an axis produce NO torque So pick points with the most forces on it! EXAMPLE: A board 6 m in length, 12 kg in mass, and of uniform mass distribution, is held by two light ropes, one on its left edge and the other 1 m away from its right edge, as shown in the first image. An 8 kg object is placed 1 m from the left end. Calculate the tension of each rope. Page 6
PRACTICE: EQUILIBRIUM WITH MULTIPLE SUPPORTS PRACTICE: A board 8 m in length, 20 kg in mass, and of uniform mass distribution, is supported by two scales placed underneath it. The left scale is placed 2 m from the left end of the board, and the right scale is placed on the board s right end. A small object 10 kg in mass is placed on the left end of the board. Calculate the reading on the left scale. BONUS: Calculate the reading on the right scale. m Page 7
CENTER OF MASS AND SIMPLE BALANCE Remember: An object s weight ALWAYS acts on its ( ). - Also: If an object has mass distribution, its is on its geometric. - An object sticking out of a supporting surface will TILT if its is located beyond the support s edge. - These are Static Equilibrium problems, BUT are solved using, which is much simpler: X = = EXAMPLE: HOW FAR CAN YOU GO ON A PLANK? EXAMPLE: A 20 kg, 10 m-long plank is supported by two small blocks, one located at its left edge and the other 3 m from its right edge. A 60 kg person walks on the plank. What is the farthest this person can get, to the right of the rightmost support, before the plank tips? Page 8
NON-UNIFORM MASS DISTRIBUTIONS Unless otherwise stated, assume a Rigid Body has UNIFORM mass distribution, so its weight acts on its. - If it does NOT have uniform mass distribution, you CANNOT assume the location of its. - In these problems, you will be given the center of mass and asked to calculate something else (or vice-versa). EXAMPLE: An 80 kg, 2 m-tall man lies horizontally on a 2 m-long board of negligible mass. Two scales are placed under the board, at its ends, as shown. If the left and right scales read 320 N and 480 N, respectively, how far from the man s head is his center of mass? Use g = 10 m/s 2 to simplify your calculations. Page 9
PRACTICE: FORCES ON A PUSH-UP PRACTICE: A 70 kg, 1.90 m man doing push-ups holds himself in place making 20 o with the floor, as shown. His feet and arms are, respectively, 1.15 m below and 0.4 m above from his center of mass. You may model him as a thin, long board, and assume his arms and feet are perpendicular to the floor. How much force does the floor apply to each of his hands? BONUS: How much force does the floor apply to each of his feet? Page 10
STATIC / COMPLETE EQUILIBRIUM IN 2D So far we ve solved Equilibrium problems that were essentially 1 dimensional: all forces acted in the same axis (X or Y). - More advanced problems have forces in 2 axes, and some will need to be. - Remember however that Torques are, so we will never need to them. EXAMPLE: A ladder of mass 10 kg (uniformly distributed) and length 4 m rests against a vertical wall while making an angle of 53 o with the horizontal, as shown. Calculate the magnitude of the: (a) Normal force at the bottom of the ladder; (b) Normal force at the top of the ladder; (c) Frictional force at the bottom of the ladder; (d) Minimum coefficient of static friction needed; (e) Total contact force at the bottom of the ladder. Page 11
PRACTICE: PERSON ON A LADDER PRACTICE: A ladder of mass 20 kg (uniformly distributed) and length 6 m rests against a vertical wall while making an angle of Θ = 60 o with the horizontal, as shown. A 50 kg girl climbs 2 m up the ladder. Calculate the magnitude of the total contact force at the bottom of the ladder (Remember: You will need first calculate the magnitude of N,BOT and f,s). Page 12
EXAMPLE: MINIMUM ANGLE AND FRICTION ON LADDER EXAMPLE: A ladder of mass M (uniformly distributed) and length L rests against a vertical wall while making an angle with the horizontal, as shown. Derive an expression for the: (a) Minimum coefficient of static friction necessary for the ladder to stay balanced at an angle of Θ; (b) Minimum angle at which the ladder can stay balanced, for a coefficient of static friction of μ,s. (c) Minimum angle at which the ladder can stay balanced, for any coefficient of friction, if there any no masses on it. Page 13
BEAM / SHELF AGAINST A WALL Some Static Equilibrium problems have shelf-like objects tensioned against a wall - In these problems, the hinge (on the wall) applies a force against the beam. - The hinge always applies a horizontal force against the. - The hinge almost always applies a force on the beam, to help hold it. - We ll assume HY is, and if you get a negative for HY, it means it was actually down which is OK! EXAMPLE: A beam 300 kg in mass and 4 m in length is held horizontally against a vertical wall by a hinge on the wall and a light cable, as shown. The cable makes an angle of 37 o with the horizontal. Calculate the: (a) Magnitude of the Tension force on the cable; (b) Magnitude and direction of the Net Force the hinge applies on the beam. Page 14
PRACTICE: BEAM SUPPORTED BY AN INCLINED ROD PRACTICE: A beam 200 kg in mass and 6 m in length is held horizontally against a wall by a hinge on the wall and a light rod underneath it, as shown. The rod makes an angle of 30 o with the wall and connects with the beam 1 m from its right edge. Calculate the angle that the Net Force of the hinge makes with the horizontal (use +/ for above/below +x axis). Page 15
EXAMPLE: BEAM SUPPORTING AN OBJECT EXAMPLE: A beam 400 kg in mass and 8 m in length is held horizontally against a wall by a hinge on the wall and a light cable, as shown. The cable makes 53 o with the horizontal and connects 2 m from the right edge of the beam. A 500 kg object hangs from the right edge of the beam. Calculate the magnitude of the net force the hinge applies on the beam. 500 kg Page 16
PRACTICE: INCLINED BEAM AGAINST A WALL PRACTICE: A beam 200 kg in mass and 4 m in length is held against a vertical wall by a hinge on the wall and a light horizontal cable, as shown. The beam makes 53 o with the wall. At the end of the beam, a second light cable holds a 100 kg object. Calculate the angle that the Net Force of the hinge makes with the horizontal (use +/ for above/below +x axis). 100 kg Page 17
EXAMPLE: INCLINED BEAM AGAINST THE FLOOR EXAMPLE: A 100 kg, 4 m-long beam is held at equilibrium by a hinge on the floor and a force you apply on its edge, as shown. The beam is held at 30 o above the horizontal, and your force is directed 50 o above the horizontal. Calculate the: (a) Magnitude of the force you apply on the beam; (b) Magnitude and direction of the Net Force the hinge applies on the beam. Page 18
PRACTICE: INCLINED BEAM AGAINST THE FLOOR PRACTICE: A 200 kg, 10 m-long beam is held at equilibrium by a hinge on the floor and a force you apply on it via a light rope connected to its edge, as shown. The beam is held at 53 o above the horizontal, and your rope makes an angle of 30 o with it. Calculate the angle that the Net Force of the hinge makes with the horizontal (use +/ for above/below +x). Page 19
CENTER OF MASS (AND CENTER OF GRAVITY) In Physics, sometimes it s useful to simplify SYSTEM of objects by replacing ALL objects with a single, equivalent object. - This single object will have mass M = and will be located at the system s : 2kg 10 m 2kg Center of Mass Equation: XCM = =. - If objects are in a 2D plane, we also have: YCM = =. EXAMPLE 1: Two masses are placed along the x-axis: mass A (10 kg) is placed at 0.0 m and mass B (20 kg) at 4.0 m. Find the Center of Mass of this system. A system s Center of GRAVITY is the same as its Center of MASS IF the gravitational field is. - Unless otherwise stated, we assume gravitational fields are constant so Center of Gravity = Center of Mass. EXAMPLE 2: Three masses are placed on an X-Y plane: mass A (10 kg) is placed at coordinates (0, 0) m, mass B (8 kg) at (0, 3) m, and mass C (6 kg) at (4, 0) m. Find the X, Y coordinates for the Center of Mass of this system. Page 20
TORQUE & STATIC EQUILIBRIUM Remember: If the on an object is, then, which we call. - However, sometimes this is not sufficient for equilibrium. For example: - So there are actually TWO conditions that are necessary for an object to have equilibrium: (1) First Condition Equilibrium (2) Second Condition Equilibrium - Static refers to the fact that and. BOTH Equilibrium - This is sometimes called Equilibrium of Rigid Bodies because we ll deal with Rigid Bodies only, no Point Masses. EXAMPLE: In all of the following, a light bar is free to rotate about a perpendicular axis through its center. The bar is not attached, so it is also free to move horizontally / vertically. All forces have the same magnitude (double arrows are a single force with double the magnitude). Ignore gravity. For each: Is the object in linear equilibrium? Is it in rotational equilibrium? [ Linear EQ Rotational ] [ Linear EQ Rotational ] [ Linear EQ Rotational ] [ Linear EQ Rotational ] [ Linear EQ Rotational ] [ Linear EQ Rotational ] Page 21
EXAMPLE: BALANCING A BAR WITH A FORCE EXAMPLE: The bar below is 4 m long and has mass 10 kg. Its mass is distributed uniformly, therefore its center of mass is located in the middle of the bar. The bar is free to rotate about a fulcrum positioned 1 m away from its left end. You want to push straight down on the left edge of the bar, to try to balance it. (a) What magnitude of force should you apply on the bar? (b) How much force does the fulcrum apply on the bar? Page 22
PRACTICE: BALANCING A COMPOSITE DISC PRACTICE: A composite disc is made out of two concentric cylinders, as shown. The inner cylinder has radius 30 cm. The outer cylinder has radius 50 cm. If you pull on a light rope attached to the edge of the outer cylinder (shown left) with 100 N, how hard must you pull on a light rope attached to the edge of the inner cylinder (shown right) so the disc does not spin? Page 23
EXAMPLE: PIN HOLDING A HORIZONTAL BAR EXAMPLE: A 20-kg, 3 m-long bar is held horizontally against a wall by a pin (shown as red). Calculate the torque the pin must provide in order to hold the bar horizontally. You may assume the bar has uniform mass distribution. Page 24