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INTRO TO TORQUE TORQUE is a twist that a Force gives an object around an axis of rotation. - For example, when you push on a door, it rotates around its hinges. - When a Force acts on an object, away from its axis, it produces a Torque on it. - Similar to how Forces cause linear acceleration, Torques cause ( ). More later! The MAGNITUDE of the Torque is given by τ = [ ] - Where r is a vector from the to the point where. - Θ is the angle between and. - Torque is max when the Force is the farthest possible and ( o ) to the r vector. - To calculate Torque, we ll use 3 steps: (1) Draw r vector (2) Figure out Θ (3) Plug numbers into equation EXAMPLE: You push/pull on a 3-m wide door with 10 N in different ways, as shown. Calculate the Torque that each force produces. F1, F4, F5 act at the edge of the door, F2 half way through it, F3 at the hinge. F5 is directed 60 o below the x-axis. F1 F4 F3 F2 F5 Page 2
EXAMPLE: TORQUE ON A FISHING POLE EXAMPLE: When a fish catches your bait, your 2 kg, 3 m long fishing pole is directed at 50 o above the +x axis. Calculate the Torque produced on your fishing pole, about an axis of rotation on your hands, if the fish pulls on it with 40 N directed at 20 o below the +x axis. Page 3
PRACTICE: TORQUE ON A HORIZONTAL LADDER PRACTICE: A 4 m-long ladder rests horizontally on a flat surface. You try to lift it up by pulling on the left end of the ladder with a force of 50 N that makes an angle of 37 o with the vertical axis. Calculate the torque that your force produces, about an axis through the other (right) end of the ladder. Page 4
EXAMPLE: FIND MAXIMUM TORQUE EXAMPLE: You must produce a torque of 100 Nm to properly tighten a given bolt using a 20-cm wrench. What is the minimum force you need to apply to the wrench to achieve this? Page 5
PRACTICE: TORQUE ON A WRENCH PRACTICE: You pull with a 100 N at the edge of a 25 cm long wrench, to tighten a bolt (gold), as shown. The angle shown is 53 o. Calculate the torque your force produces on the wrench, about an axis perpendicular to it and through the bolt. Page 6
NET TORQUE AND THE SIGN OF TORQUE The SIGN of Torque depends on which direction the Force causes the object to spin CW is ; CCW is - If multiple Torques are produced on an object, we can calculate the NET Torque τnet = - Torques are, so we use simple addition (not vector addition) to find Net Torque. EXAMPLE: Two forces act on the same 3-m wide door, as shown. F1 acts on the center of the door, and F2 is directed 30 o above the x-axis. Calculate the Net Torque produced on the door. Use signs (+/ ) to indicate the direction of the Torques. F2 = 50 N F1 = 50 N Page 7
PRACTICE: NET TORQUE / FORCES ON A BAR PRACTICE: A 2-m long bar is free to rotate about an axis located 0.7 m from one of its ends. Two forces act on the bar, F1 = 100 N and F2 = 200 N, and both make 30 o with the bar. Find the Net Torque on the bar. Use +/ to indicate direction. F2 F1 Page 8
TORQUE DUE TO WEIGHT An object s weight ALWAYS acts on its ( ). - If an object has mass distribution, its is on its geometric. EXAMPLE: A 20 kg, 4 m long, cylindrical rod has one of its ends fixed to an axis that is mounted on the floor, as shown. The rod is adjusted to point 37 o above the horizontal. Suppose you have mass 80 kg, and stand on the other end of the rod. Calculate the Net Torque that is produced on the rod, about its axis, due to TWO weight forces acting on it. You may assume the rod has uniform mass distribution and is fixed in place, so it does not move or rotate. Page 9
PRACTICE: NET TORQUE / KIDS ON A SEESAW PRACTICE: Two kids play on a seesaw that has mass 20 kg, length 3 m, and its fulcrum at its mid-point. The seesaw is originally horizontal, when the kids sit at the edge of opposite ends (m,left = 25 kg, m,right = 30 kg). Calculate the Net Torque from the 3 weights acting on the seesaw, immediately after the kids sit (simultaneously) on their respective places. Page 10
PRACTICE: NET TORQUE / HOLDING BARBELL PRACTICE: A guy standing straight up stretches out his arm horizontally while holding a 60 lb (27.2 kg) barbell. His arm is 64 cm long and weighs 45 N. Calculate the Net Torque that the barbell and the weight of his arm produce about his shoulder. You may assume that his arm has uniform mass distribution. Page 11
TORQUE ON DISCS / PULLEYS Problems of Torques on discs are common, and will be useful later. - Note that what matters is r (axis to force), not radius R EXAMPLE: Two masses (m1 = 4 kg, m2 = 5 kg) are connected by a light string which is passed through the edge of a solid cylinder (m3 = 10 kg, radius = 3 m), as shown. The system is free to rotate about an axis perpendicular to the cylinder and through its center. Calculate the Net Torque produced on the cylinder, about its central axis, when you release the blocks. m1 m2 Page 12
EXAMPLE: TORQUES ON A DISC EXAMPLE: The composite disc below is free to rotate about a fixed axis, perpendicular to it and through its center. All forces are 100 N, and all angles are 37 o. The dotted lines are either exactly parallel or exactly perpendicular to each other. The inner (darker) and outer (lighter) discs have radii 3 m and 5 m, respectively. Calculate the Net Torque produced on the composite disc, about an axis perpendicular to it and through its center. Use +/ to indicate direction. Page 13
PRACTICE: TORQUES ON A DISC PRACTICE: The composite disc below is free to rotate about a fixed axis, perpendicular to it and through its center. All forces are 100 N, and all angles are 37 o. The dotted lines are either exactly parallel or exactly perpendicular to each other. The inner (darker) and outer (lighter) discs have radii 3 m and 5 m, respectively. Calculate the Net Torque produced on the composite disc, about an axis perpendicular to it and through its center. Use +/ to indicate direction. Page 14
PRACTICE: TORQUES ON A SQUARE PRACTICE: A square with sides 4 m long is free to rotate around an axis perpendicular to its face and through its center. All forces shown are 100 N and act simultaneously on the square. The angle shown is 30 o. Calculate the Net Torque that the forces produce on the square, about its axis of rotation. Page 15
TORQUE & ACCELERATION (ROTATIONAL DYNAMICS) When a Force causes rotation, it produces a Torque. Think of TORQUE as the equivalent of FORCE! FORCE (F) TORQUE (τ) - Causes linear acceleration ( ) - Relationship between F & - Remember: This is Newton s! - Quantity of Inertia (resistance to a) - Force & Acceleration - Causes angular/rotational acceleration ( ) - Relationship between τ & - of Newton s! - Quantity of Inertia (resistance to α) - τ & α EXAMPLE: A solid disc of mass M = 100 kg and radius R = 2 m is free to rotate around a fixed axis that is perpendicular to it, runs through its center, and is frictionless. You push tangentially on the disc with a constant force F = 50 N, as shown. (a) Derive an expression for the angular acceleration that the disc experiences. (b) Calculate this angular acceleration. Page 16
PRACTICE: TORQUE & ACCELERATION / WEIRD SHAPE (PIANO) PRACTICE: Suppose that piano has a long, thin bar ran through it (totally random), shown below as the vertical red line, so that it is free to rotate about a vertical axis through the bar. You push the piano with a horizontal 100 N (blue arrow), causing it to spin about its vertical axis with 0.3 rad/s 2. Your force acts at a distance of 1.1 m from the bar, and is perpendicular to a line connecting it to the bar (green dotted line). What is the piano s moment of inertia about its vertical axis? Page 17
TORQUE & ACCELERATION / POINT MASS Most Torque problems involve Shapes/Rigid Bodies, but Torque works just the same for Point Masses! EXAMPLE: You spin a small rock of mass M = 2 kg at the end of a light string of length L = 3 m. (a) What Net Torque is needed to give the rock an acceleration of 4 rad/s 2? (b) Calculate its tangential acceleration while it spins with 4 rad/s 2. Page 18
HOW TO SOLVE: TORQUE VS. CONSERVATION OF ENERGY Remember: Some Linear motion problems can be solved with ΣF=ma and Motion Equations OR Conservation of Energy: - For example, there are two ways to find the velocity of the block at the bottom of the plane: m - Likewise, some Rotational motion problems can be solved with Στ =Iα and/or Motion or Conservation of Energy. - Depending on what you re being asked and what you re being given, one method is better than the other: - Generally, you will use Στ =Iα to solve problems asking for (or giving) or. - Use Conservation of Energy to solve problems asking for (or giving) or. - ALWAYS use Motion Equations if looking for ( ) or need it to solve a problem. - Sometimes you may be asked to use a specific method, in which case you have no choice :( Two questions may look almost identical, but require very different methods to solve. For example: - A yo-yo spins around itself as it falls. Find its acceleration after dropping 2 m - A yo-yo spins around itself as it falls. Calculate its speed after dropping 2 m - A yo-yo spins around itself as it falls. How long does it take to drop 2 m? Page 19