ReviewSolutions - Physics 8A Midterm 2 Review Solutions...

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Physics 8A Midterm 2 Review Solutions Problem 1 1. θ R M mg T f s x y N Figure 1: Free Body Diagram 2. The cylinder is in static equilibrium, so the Σ ~ F = m~a = 0 and Σ τ = = 0. Choose coordinates as indicated on fgure 1, with counterclockwise as positive ±or torque. Σ F x = f + T cosθ - Mgsinθ = 0 (1) Σ F y = N - T sinθ - Mgcosθ = 0 (2) Σ τ = fR - T R = 0 (3) We can use (3) and (1) to fnd the magnitude o± the ±riction. From (3), we get f = T . Pluging into (1), We get Σ F x = f (1 + cosθ ) - Mgsinθ = 0 (4) which simplifes to f = Mgsinθ (1 + cosθ ) (5) 3. To fnd the minimum μ s that will prevent the block ±rom slipping, use the ±act that f μ s N . so the minumum will be when f = μ s N . The rest o± is just a bunch o± algebra, which is probably more involved than you will see on the midterm. Don’t worry too much i± you couldn’t get it to simpli±y all the way. Plug N = f μ s and T = f into (2) N - T sinθ - Mgcosθ = f ( 1 μ s - sinθ ) - Mgcosθ = 0 (6) Using (5), we get Mgsinθ (1 + cosθ ) ( 1 μ s - sinθ ) - Mgcosθ = 0 (7) or Mgsinθ μ s - Mgsin 2 θ - (1 + cosθ ) Mgcosθ = 0 (8)
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multiplying by μ s and dividing by Mg, we get sinθ - μ s sin 2 θ - μ s cosθ - μ s cos 2 θ = 0 (9) using the fact that sin 2 θ + cos 2 θ = 1, we get sinθ - μ s - μ s cosθ = 0 (10) and solving for μ s we get μ s = sinθ 1 + cosθ = 0 (11) Problem 2 1. ρ ρ l a c F mg b F ext Cube Figure 2: the forces acting on the block are from gravity and the bouyancy force. A downward external force must be applied to keep it submerged. To ±nd the minimum force required us Σ ~ F = 0, which tells us F ext = F b - Mg = ρ l V disp g - ρ c V c g = ρ l a 3 g - ρ c a 3 g = a 3 g ( ρ l - ρ c ) (12) The external force is now coming from a volume of material being attached to it. From the free body diagram of the material, we see that T = m m g - F bm = m m g - ρ l V disp by m g (13) so using (12) with F ext = T and using the fact that m m = V m ρ m , we get m m g - ρ l V dispbym g = V m ρ m - ρ l V disp by m ga 3 g ( ρ l - ρ c ) (14) and solving for
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This note was uploaded on 09/20/2008 for the course PHYSICS 8A taught by Professor Jacobsen during the Fall '07 term at University of California, Berkeley.

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ReviewSolutions - Physics 8A Midterm 2 Review Solutions...

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