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TORQUE, CENTER OF MASS, CENTER OF GRAVITY Semester I 2010/2011 Motion in which an entire object moves is called translation Motion in which an object spins is called rotation The point or line about which an object turns is its center of rotation The LEVER ARM, d, is the perpendicular distance from the axis of rotation to the line of action of the force. d = L sin Φ FULCRUM - the point of support, or axis, about which a lever may be made to rotate How does force create rotation ?? A torque is an action that causes objects to rotate. Torque is not the same thing as force. Torque rF F r TORQUE is the cross product between the distance vector (the distance from the pivot point to the point where force is applied) and the force vector magnitude : rF sin rF sin To make an object rotate, a force must be applied in the right place. 0 0 Torque rF magnitude : rF sin Unit Nm Consider force required to open door. Is it easier to open the door by pushing/pulling away from hinge or close to hinge? CLOSE? AWAY? The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation. What property of the applied force causes the door to open? rF sin Using the concept of torque, explain why the “easy-off” cap is easier to unscrew than the normal cap. Some Physics Quantities Vector - quantity with both magnitude (size) and direction Scalar - quantity with magnitude only Vectors: Scalars: •Velocity •Energy • Force •Work Cross product The cross product of two vectors a and b is defined as c ab rF whose magnitude is c ab sin rF sin where is the angle (< 180o) between a and b, whose direction is perpendicular to both a and b in the sense of the right-hand rule. Cross product (right-hand rule ) c ab Use the right-hand rule to determine the direction of the resulting vector in a cross product. Hold your right hand in front of you so that the thumb is pointed up, the index finger is pointed away from you and the middle finger is pointed to your left. The index finger shows the direction of vector A, the middle finger shows the direction of vector B and the thumb shows the direction of the vector from the cross product A x B. • A torque ( a vector quantity) that tends o produce a counterclockwise rotation is considered positive. • A torque that tends to produce a clockwise rotation is negative. By convention, the sign of torque is: <0 clockwise >0 counter-clockwise A force of 50 newtons is applied to a wrench that is 30 centimeters long. Calculate the torque if the force is applied perpendicular to the wrench so the lever arm is 30 cm rF sin (0.3m)(50 N ) sin 90 o rF sin (0.3m)(50m)(1) 15Nm For the same force, you get more torque with a bigger wrench the job is easier! The torque can be increased by applying the force at right angles to the lever arm or by extending the lever arm. For your arm, leg or any body part to move the appropriate muscles and bones must work together as a series of levers. A lever comprises of three components : Fulcrum or pivot - the point about which the lever rotates Load - the force applied by the lever system Effort - the force applied by the user of the lever system The way in which a lever will operate is dependent on the type of lever. Levers are important in human motion because the human body is a system of levers. Our joints are axes of rotation (fulcrums) and our bones are the levers. Forces to move the levers are provided by our muscles, gravity, and external forces. Three Classes of Levers First Class - fulcrum between Input and output Second Class – output between fulcrum and input Third Class – input between fulcrum and output First Class Second Class Third Class Mechanical Equilibrium • To ensure that an object does not accelerate or rotate two conditions must be met: • net force = 0 • net torque = 0 Mechanical Equilibrium • First Condition of Equilibrium • The net external force must be zero F 0 Fx 0 and Fy 0 – This is a statement of translational equilibrium • Second Condition of Equilibrium • The net external torque must be zero 0 • This is a statement of rotational equilibrium SEESAW 2m 2m 200N 400N Unbalanced torques balanced torques 2m 1m 200N 400N For equilibrium to exists, torques must add to zero. The torques applied by the 2 kids are equal and opposite, so the see-saw doesn’t move. Balanced torques happen on a seesaw when two children of different weights sit at different positions We know that the heavier kid needs to sit closer to the center of the see-saw. Why? In order for the see-saw to balance, there must be no rotation. This means that there cannot be any unbalanced torques acting on the see-saw. The heavier kid has more weight (force) so must have a smaller lever arm in order to make the product of force and lever arm (torque) equal that of the smaller kid sitting further from the axis. balanced torques • Direction of rotation of applied torque is very important (i.e. clockwise or anticlockwise). • Torques can add or oppose each other. • If two opposing torques are of equal magnitude they will cancel one another to create a balanced system. r1 r2 W1 = m1.g W2 = m2.g Torque = Fr W1.r1 = W2.r2 or m1.g.r1 = m2. g.r2 Thus at balance: m1.r1 = m2.r2 (This is the principle of weighing scales.) If an object is supported at one special point, it will balance. This point is called the center of gravity . •For SYMMETRICAL objects, such as a meter stick, the CG is located at the actual center. • For ASYMMETRICAL objects, such as a hammer, the CG is located away from the center, closer to the hammer head (the heavier head) Center of Mass (CM) • An object can be divided into many small particles – Each particle will have a specific mass and specific coordinates • The x coordinate of the center of mass will be m x i i xCM i m i i • Similar expressions can be found for the y and z A system can often be well represented coordinates as though its mass were concentrated at single point, the center of mass Center of Mass examples CM CM CM CM for the Human Body The location of the center of mass of the leg (circled) will depend on the position of the leg. High jumpers have developed a technique where their CM actually passes under the bar as they go over it. This allows them to clear higher bars. CM Center of Gravity • All the various gravitational forces acting on all the various mass elements are equivalent to a single gravitational force acting through a single point called the center of gravity. The center of gravity is the point where the gravitational force can be considered to act. It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object. A force applied at the CG of an object results in straight-line acceleration, but no rotation. A force applied away from the CG will have a “lever arm”, which produces a torque. The result is straight-line acceleration and rotation. Why things fall over • Every object has a special point called the center of gravity (CG). • if the center of gravity is supported, the object will not fall over. Touch your toes while standing against a wall Why things fall over

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posted: | 4/11/2011 |

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