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Rotational Mechanics

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**Torque The ability of a force to rotate an object around some axis**

Produces rotation Produced when a force is applied with leverage Longer the lever = more torque More force = more torque Ex: Door

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**More Torque τ = Fd Torque = Force x lever arm Units: N x m = Nm**

Lever arm: distance from the turning axis to the point of contact Force is always perpendicular to lever arm. Torque = Force x lever arm τ = Fd Units: N x m = Nm

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**Torque also depends on the angle between force and lever arm**

τ = Fd (sin θ) Perpendicular = most torque Increase angle = less torque Increase angle more = least torque

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**Sign of Torque Counterclockwise rotation = (+) torque**

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Problems A force of 50N is applied perpendicular to the door, .5m from the hinges. How much torque is produced? What if the force were applied .1 m from the hinges? If you cannot exert enough torque with a wrench to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle?

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Example Problem Ned tightens a bolt in his car engine by exerting 12 N of force on his wrench at a distance of 0.40 m from the fulcrum. How much torque must Ned produce to turn the bolt?

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Sample Problem 8A A basketball is being pushed by two players during tip-off. One player exerts a downward force of 11 N at a distance of 7.0 cm from the axis of rotation. The second player applies an upward force of 15 N at a perpendicular distance of 14 cm from the axis of rotation. Find the net torque acting on the ball.

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Balanced Torques Ex: on a seesaw, balance can be achieved if clockwise torque = counter clockwise torque If a meter stick is balanced in the center A 20 N block is attached to the 80cm mark, and another block is attached to the 10cm mark. What is the weight of the second block?

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**Balancing Torque Problem**

Mabel and Maude are seesawing on the school playground and decide to see if they can move to the correct location to make the seesaw balance. Mabel weighs 400. N and she sits 2.00 m from the fulcrum of the seesaw. Where should 450. N Maude sit to balance the seesaw?

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**Center of Mass Average position of the mass of an object**

Object will rotate around this point is gravity is the only force acting on it. When working problems – all mass can be considered to be concentrated here For regular shapes – CoM is at the center of the object Center of Gravity – average position of weight of an object An object will fall over if the center of gravity extends beyond the area of support. Falling over C o G outside the object Usually, the center of mass and center of gravity are at the same point.

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Rotational Inertia The resistance of an object to changes in its rotational motion. AKA: moment of inertia An object rotating about an axis tends to keep rotating about that axis.

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Moment of Inertia Torque is required to change the rotational state of motion of an object. Rotational inertia depends on mass and on the distribution of the mass. The greater the distance between the axis and the majority of the mass, the greater the rotational inertia. Ex: Bat

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**Formulas for Rotational Inertia**

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Rolling Which will roll down an incline with greater acceleration, a hollow cylinder or a solid cylinder if both have the same mass and diameter? Solid cylinder Why? Smaller cylinder has less rotational inertia.

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**Rotational Equilibrium**

Equilibrium requires zero net force and zero net torque Translational Equilibrium ΣF = 0 Rotational Equilibrium ΣΤ = 0

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Sample Problem 8B A uniform 5.00 m long horizontal beam that weighs 315 N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53o with the horizontal, and a 545 N person is standing 1.50 m from the pin. Find the force in the cable, FT, and the force exerted on the beam by the wall, R, if the beam is in equilibrium.

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**Newton’s 2nd Law of Rotation**

Tnet = Iα Net torque = moment of inertia x angular accel. A bicycle tire of radius .330 m and mass 1.50 kg is rotating at 98.7 rad/s. What torque is needed to stop the tire in 2.00 s?

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Sample Problem 8C A student tosses a dart using only the rotation of her forearm to accelerate the dart. The forearm rotates in a vertical plane about an axis at the elbow joint. The forearm and dart have a combined moment of inertia of kg x m2 about the axis, and the length of the forearm is 0.26 m. If the dart has a tangential acceleration of 45 m/s2 just before it is released, what is the net torque on the arm and dart?

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**Angular Momentum Angular momentum = inertia of rotation**

Angular momentum = rotational inertia x rotational velocity Angular momentum L = x Angular momentum always acts outward along the axis Angular momentum = mvr

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**Law of Conservation of Angular Momentum**

If no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant.

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Example A merry-go-round rotates at the rate of 0.30 rad/s w/an 80. kg man standing at a point 2.0 m from the axis of rotation. If the man walks to a point 1.0 m form the center, what will be the new angular speed? Merry-go-round = solid cylinder, m = 6.5 x 102 kg, r=2.0 m

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Sample Problem 8D A 65 kg student is spinning on a merry-go-round that has a mass of 5.25 x 102 kg and a radius of 2.00 m. She walks from the edge of the merry-go-round toward the center. If the angular speed of the merry-go-round is initially 0.20 rad/s, what is the angular speed when the student reaches a point 0.50 m from the center.

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**Homework, yea!!!! Practice problems**

8D: 3, 4, & 5 Stop complaining, it’s only 6 problems

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**Rotational Kinetic Energy**

Just as with an object moving in a straight line, a rotating object also possess kinetic energy KErot = ½ I ω2 This causes the conservation of energy equation to change to (PE + KEtrans +KErot)i = (PE + KEtrans +KErot)f We call this Mechanical Energy ME = KEtrans + KErot + Peg ME = ½ mv2 + ½ Iω2 + mgh

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Example A 1.5 kg bicycle tire of radius .33m starts from rest and rolls down a hill 14.8m high. What is the translational speed when it reaches the bottom of the hill?

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Sample Problem 8E A solid ball with a mass of 4.10 kg and a radius of m starts from rest at a height of 2.00 m and rolls down a 30.0o slope. What is the translational speed of the ball when it leaves the incline?

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Homework Practice 8E (all)

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Simple Machines Machine – any device that transmits or modifies force, usually by changing the force applied to an object. Types of simple machines Lever Inclined Plane Wheel & Axle Pulley Wedge Screw

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**Mechanical Advantage MA compares output force to input force**

MA = Fout/Fin Fout = resistance Fin = effort The MA can also be compared to the distance moved – like in a lever or incline MA = din/dout din = effort dout = resistance

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**In the absence of friction, energy is conserved in a machine**

The force applied may be decreased, if the distance it is applied is increased. The product of the 2 remains constant In reality, no machine exists without friction We can calculate the efficiency of a machine by.. Eff = Wout/Win Wout = resistance Win = effort

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Example You are pushing a block up a ramp that is 3.0 m high and 6.0 m long. It is a 30.o ramp. The block has a mass of 5.0 kg. It is takes you 37 N of force to push it up the incline. Draw this picture. What is the Effort Force? What is the Resistance Force? What is the Effort Distance? What is the Resistance Distance? What is the Mechanical Advantage? What is the Efficiency of the machine?

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Homework Section Review (all)

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**Let’s try these Starting on page 306:**

#12, 14, 18, 21, 26, 31, 35, & 45

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