qz6sol_6942s11 - edge E i = E f ⇒ mg(Lsin θ =(1/2)mv...

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TA: Tomoyuki Nakayama Monday, March 14, 2011 PHY 2048: Physic 1, Discussion Section 6942 Quiz 6 (Homework Set #8) Name: UFID: Formula sheets are not allowed. Do not store equations in your calculator. You have to solve problems on your own; memorizing final algebraic expressions from homework assignments and just plugging numbers into them will not give you full credit. Leave all your work. ________________________________________________________________________________ In the figure, a solid uniform sphere of radius 5.00 cm and mass 2.00 kg starts from rest and rolls without slipping a distance L = 6.00 m down a roof that is inclined at angle θ = 40°. The rotational inertia of a solid sphere is I = (2/5)mr 2 . a) What is the angular speed of the cylinder about its center as it leaves the roof? When the sphere rolls down the roof, the mechanical energy of the sphere is conserved. We apply the energy conservation equation to the initial position and the edge of the roof and solve the equation for the angular speed at the
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Unformatted text preview: edge. E i = E f ⇒ mg(Lsin θ ) = (1/2)mv 2 + (1/2)I ω 2 mgLsin θ = (1/2)m(r 2 ω 2 ) + (1/2)(2/5)mr 2 ω 2 = (7/10)mr 2 ω 2 ω = √ (10gLsin θ /7r 2 ) = 147 rad/s b) The roof's edge is at height H = 5.00 m. How far horizontally from the roof's edge does the cylinder hit the level ground? After the sphere leaves the roof, it undergoes a projectile motion. The initial speed of the projectile motion is the speed of the sphere at the edge of the roof: v = r ω = 7.35 m/s At the edge, the sphere is moving at an angle of 40º below the horizontal. x and y components of the initial velocity of the projectile motion are v 0x = v =cos(-40º) = 5.63 m/s, v 0y = v sin(-40º) = -4.72 m/s When the sphere hits the ground, the vertical displacement of the sphere is –H. The flight time is -H = v 0y t – (1/2)gt 2 t = (-v 0y + √ (v 0y 2 + 2gH))/g = 0.637 s During this time interval, the horizontal displacement of the sphere is Δ x = v 0x t = 3.59 m...
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