YF_ISM_09

YF_ISM_09 - ROTATION OF RIGID BODIES 9 9.1. 9.2. IDENTIFY:...

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9-1 R OTATION OF R IGID B ODIES 9.1. IDENTIFY: sr θ = , with in radians. SET UP: rad 180 π = ° . EXECUTE: (a) 1.50 m 0.600 rad 34.4 2.50 m s r == = = ° (b) 14.0 cm 6.27 cm (128 )( rad /180 ) s r θπ = °° (c) (1.50 m)(0.700 rad) 1.05 m = EVALUATE: An angle is the ratio of two lengths and is dimensionless. But, when = is used, must be in radians. Or, if / = is used to calculate , the calculation gives in radians. 9.2. IDENTIFY: 0 t θθ ω −= , since the angular velocity is constant. SET UP: 1 rpm (2 /60) rad/s = . EXECUTE: (a) (1900)(2 rad /60 s) 199 rad/s ωπ (b) 35 (35 )( /180 ) 0.611 rad ° . 3 0 0.611 rad 3.1 10 s 199 rad/s t θθ ω = × EVALUATE: In 0 t = we must use the same angular measure (radians, degrees or revolutions) for both 0 and . 9.3. IDENTIFY: () z z d t dt α = . Writing Eq.(2.16) in terms of angular quantities gives 2 1 t z t dt . SET UP: 1 nn d tn t dt = and 1 1 1 td t t n + = + EXECUTE: (a) A must have units of rad/s and B must have units of 3 rad/s . (b) 3 ( ) 2 (3.00 rad/s ) z tB t t . (i) For 0 t = , 0 z = . (ii) For 5.00 s t = , 2 15.0 rad/s z = . (c) 2 1 23 3 1 21 3 ( ) ( ) t t AB t A t t B + = − + . For 1 0 t = and 2 2.00 s t = , 33 1 3 (2.75 rad/s)(2.00 s) (1.50 rad/s )(2.00 s) 9.50 rad + = . EVALUATE: Both z and z are positive and the angular speed is increasing. 9.4. IDENTIFY: / zz dd t αω = . av- z z t Δ = Δ . SET UP: 2 () 2 d tt dt = EXECUTE: (a) 3 2 ( 1 .60 rads). z z d ω α t β dt (b) 32 (3.0 s) ( 1.60 rad s )(3.0 s) 4.80 rad s . z α =− = 2. av- (3.0 s) (0) 2.20 rad s 5.00 rad s 2.40 rad s , 3.0 s 3.0 s z ωω α −− = which is half as large (in magnitude) as the acceleration at 3.0 s. t = EVALUATE: z t increases linearly with time, so av- (0) (3.0 s) 2 z αα + = . (0) 0 z = . 9
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9-2 Chapter 9 9.5. IDENTIFY and SET UP: Use Eq.(9.3) to calculate the angular velocity and Eq.(9.2) to calculate the average angular velocity for the specified time interval. EXECUTE: 3 ; tt θ γβ =+ 0.400 rad/s, γ = 3 0.0120 rad/s β = (a) 2 3 z d t dt ω == + (b) At 0, t = 0.400 rad/s z ωγ (c) At 5.00 s, t = 32 0.400 rad/s 3(0.0120 rad/s )(5.00 s) 1.30 rad/s z = 21 av- z ttt θθθ Δ− For 1 0, t = 1 0. = For 2 5.00 s, = 33 2 (0.400 rad/s)(5.00 s) (0.012 rad/s )(5.00 s) 3.50 rad = So av- 3.50 rad 0 0.700 rad/s. 5.00 s 0 z EVALUATE: The average of the instantaneous angular velocities at the beginning and end of the time interval is 1 2 (0.400 rad/s 1.30 rad/s) 0.850 rad/s. += This is larger than av- , z because ( ) z t is increasing faster than linearly. 9.6. IDENTIFY: () z d t dt = . z z d t dt α = . av- z t Δ = Δ . SET UP: 23 2 (250 rad s) (40.0 rad s ) (4.50 rad s ) z ω =− . z (40.0 rad s ) (9.00 rad s ) α t . EXECUTE: (a) Setting 0 z ω = results in a quadratic in t . The only positive root is 4.23 s t = . (b) At 4.23 s t = , 2 78.1 rad s . z α (c) At 4.23 s t = , 586 rad 93.3 rev θ . (d) At 0 t = , 250 rad/s z ω = . (e) av- 586 rad 138 rad s. 4.23 s z EVALUATE: Between 0 t = and 4.23 s t = , z decreases from 250 rad/s to zero. z is not linear in t , so av- z is not midway between the values of z at the beginning and end of the interval. 9.7. IDENTIFY: z d t dt = . z z d t dt = . Use the values of and z at 0 t = and z at 1.50 s to calculate a , b , and c . SET UP: 1 nn d tn t dt = EXECUTE: (a) 2 3 z tbc t . ( ) 6 z tc t . At 0 t = , /4 rad a θπ and 2.00 rad/s z b . At 1.50 s t = , 2 6 (1.50 s) 1.25 rad/s z c = and 3 0.139 rad/s c .
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YF_ISM_09 - ROTATION OF RIGID BODIES 9 9.1. 9.2. IDENTIFY:...

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