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Chapter 7 • Rotational Motion Slide 7-3 Curvilinear coordinates = © 2015 Pearson Education, Inc. 1 2 ∆ = + 2 2 = 2 + 2∆ 1 2 2 2 = ∆ = + + 2∆ 2 © 2015 Pearson Education, Inc. = © 2015 Pearson Education, Inc. ∆ = ∆ This slope is © 2015 Pearson Education, Inc. Checking Understanding Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. Slide 7-13 Answer Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. All points on the turntable rotate through the same angle in the same time. All points have the same period. Slide 7-14 Angular acceleration α measures how rapidly the angular velocity is changing: t = t Tangential acceleration Slide 7-17 Slide 7-18 Checking Understanding Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The speed of A is twice that of B. B. The speed of A equals that of B. C. The speed of A is half that of B. Slide 7-15 Answer Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The speed of A is twice that of B. B. The speed of A equals that of B. C. The speed of A is half that of B. v r Twice the radius means twice the speed Slide 7-16 © 2015 Pearson Education, Inc. © 2015 Pearson Education, Inc. © 2015 Pearson Education, Inc. © 2015 Pearson Education, Inc. Center of mass follows original trajectory © 2015 Pearson Education, Inc. Slide 7-19 Example Problem A high-speed drill rotating CCW takes 2.5 s to speed up to 2400 rpm. A. What is the drill’s angular acceleration? B. How many revolutions does it make as it reaches top speed? 1 2 ∆ = + 2 2 = 2 + 2∆ 2∆ = 2 2 = ∆ 2 Slide 7-21 © 2015 Pearson Education, Inc. Slide 7-22 The speed is changing © 2015 Pearson Education, Inc. Center of Gravity = Slide 7-29 Calculating the Center-of-Gravity Position Slide 7-30 1 1 + 2 2 5kg ∙ m 1 cg = = = m 1 + 2 15kg 3 © 2015 Pearson Education, Inc. © 2015 Pearson Education, Inc. = sin © 2015 Pearson Education, Inc. Angle between lever arm and force Checking Understanding Which point could be the center of gravity of this L-shaped piece? Slide 7-32 Answer Which point could be the center of gravity of this L-shaped piece? (a) Slide 7-33 Interpreting Torque Torque is due to the component of the force perpendicular to the radial line. rF rF sin Slide 7-25 Signs and Strengths of the Torque Slide 7-27 The four forces below are equal in magnitude. Which force would be most effective in opening the door? 1 2 3 4 Either 1 or 3 98% 2% $$ 4 3 0% $$ 1 or $$ er Ei th 3 0% $$ 2 $$ 1 0% $$ A. B. C. D. E. Example torque Problem = sin Revolutionaries attempt to pull down a statue of the Great Leader by pulling on a rope tied to the top of his head. The statue is 17 m tall, and they pull with a force of 4200 N at an angle of 65° to the horizontal. What is the torque they exert on the statue? If they are standing to the right of the statue, is the torque positive or negative? = 17m ∙ 4200N ∙ sin ° ≠ 65 = 17m ∙ 4200N ∙ sin 25° 17 m Negative torque, but why? Rotating it in the CW direction r = 17m 65° pivot F = 4200 N Slide 7-28 Which force vector on point P would keep the wheel from spinning? 98% 0% E 2% D 0% C A C D E A A. B. C. D. = sin = sin = 1.6m 3.2kg m ° −9.8 2 sin 65 s © 2015 Pearson Education, Inc. Which torques are equal? 98% A. B = C = D = E only B. A = B and C = D = E C. None are equal D. B = E and C = D 2% =D C nd Ea = B No ne C d an B = A ar e = D = eq ua l y Eo nl = D = C = B 0% E 0% What is the Net Torque is exerted by the gymnast about an axis through the rings? © 2015 Pearson Education, Inc. =0 1 1 2 2 © 2015 Pearson Education, Inc. Reading Quiz 2. Which factor does the torque on an object not depend on? A. B. C. D. The magnitude of the applied force. The object’s angular velocity. The angle at which the force is applied. The distance from the axis to the point at which the force is applied. Slide 7-7 Answer 2. Which factor does the torque on an object not depend on? A. B. C. D. The magnitude of the applied force. The object’s angular velocity. The angle at which the force is applied. The distance from the axis to the point at which the force is applied. Slide 7-8 Example Problem An object consists of the three balls shown, connected by massless rods. Find the x- and y-positions of the object’s center of gravity. 1 1 + 2 2 + 3 3 cg = 1 + 2 + 3 1 1 + 2 2 + 3 3 cg = 1 + 2 + 3 Slide 7-31 An object consists of the three balls shown, connected by massless rods. Find the x- and y-positions of the object’s center of gravity. 1 0 + 2 0 + 3 1m 2kg ∙ m 1 cg = = = m 1 + 2 + 3 4kg 2 1 1m + 2 0 + 3 0 1kg ∙ m 1 cg = = = m 1 + 2 + 3 4kg 4 The center of mass for these 3 bodies Slide 7-31 Newton’s Second Law for Rotation /I I = moment of inertia. Objects with larger moments of inertia are harder to get rotating. I mi ri 2 Slide 7-34 Rotational and Linear Dynamics Compared Slide 7-36 Which moment of inertia is greatest? 25% 25% 25% 25% D C B A B C D A A. B. C. D. 20% 20% 20% E B A B C D E A A. B. C. D. E. 20% D 20% C Which force vector applied to point P will stop this rolling ball? © 2015 Pearson Education, Inc. Which red vector is your best bet for getting this bolt as tight as possible? 25% 25% 25% D C B A B C D A A. B. C. D. 25% Reading Quiz 1. Moment of inertia is A. B. C. D. the rotational equivalent of mass. the point at which all forces appear to act. the time at which inertia occurs. an alternative term for moment arm. Slide 7-5 Answer 1. Moment of inertia is A. B. C. D. the rotational equivalent of mass. the point at which all forces appear to act. the time at which inertia occurs. an alternative term for moment arm. Slide 7-6 ≠0 = What happens to these masses when you let go? © 2015 Pearson Education, Inc. What happens to this pulley system? 25% 25% 25% ac ce le fo ra rc te e st of .. gr a vi Th ty em on as th sm e ... ov es up w ar d at a. .. Th e Th e 10 N fo rc e It do es n ot m ov e 25% A. It does not move B. The 10N force accelerates the mass upward C. The force of gravity on the mass results in a net force upward D. The mass moves upward at a constant speed Starting from rest, how long does it take to hit the ground? 1 = 1 + 1 = 1 1 1 2 = 2 + 2 = 2 2 Newton’s Third 1 1 = −2 © 2015 Pearson Education, Inc. 2 Starting from rest, how long does it take to hit the ground? 2 = 1 = 1 + 1 = 1 2 − 1 = 2 1 Get rid of 1 by solving for it and substituting into the other equation m = −3.27 2 s 1 2 ∆ = + 2 = 2∆ 1 2 = .78s © 2015 Pearson Education, Inc. 2 Reading Quiz 4. A net torque applied to an object causes A. a linear acceleration of the object. B. the object to rotate at a constant rate. C. the angular velocity of the object to change. D. the moment of inertia of the object to change. Slide 7-11 Answer 4. A net torque applied to an object causes A. a linear acceleration of the object. B. the object to rotate at a constant rate. C. the angular velocity of the object to change. D. the moment of inertia of the object to change. Slide 7-12 Draw the normal force for the wheel against the break Draw the frictional force from the break © 2015 Pearson Education, Inc. Is this beam balanced? A. B. C. D. Yes No, it will spin CW No, it will spin CCW Not enough information W CC CW in gh ill No te no u tw No ,i tw ill sp sp in in Ye s No ,i fo rm at io n 25% 25% 25% 25% 1 = 2 2 = 1 + 2 = © 2015 Pearson Education, Inc. 1 = 2 2 4 kg .02m ∙ 20N − .02m ∙ 30N = .0008 kg ∙ m2 © 2015 Pearson Education, Inc. Additional Example Problem A baseball bat has a mass of 0.82 kg and is 0.86 m long. It’s held vertically and then allowed to fall. What is the bat’s angular acceleration when it has reached 20° from the vertical? (Model the bat as a uniform cylinder). Slide 7-43