Question

6) In the movie "Raiders of the Lost Ark", our hero Indiana Jones needs to outrun a giant spherical boulder that rolls downhill towards him. Assume the sphere is uniform. has mass M and radius R and starts from rest at the top of an incline of angle 8. It rolls without slipping, and its center of mass drops a vertical distance h. a) Derive an expression for the velocity of the sphere at the bottom in terms of g and h. b) What is the minimum speed that Indiana Jones needs to run if h= 5.00 m? (plug into above expression) m/s c) What % of the sphere's total kinetic energy is rotational kinetic energy? d) Assume it takes t seconds for the rolling sphere to make the first complete revolution Compute the fraction of time to make the second revolution compared to the first, or (t2-1) . (3 sig figs) finartin of a two-density evlinder about an axis through its middle and

Answer 1

Give that mass of the sphere in and radius ① af the sphere is R and height of inclined plane h with inclination o.. Weinrest) 9) To find expression ef velocity out the bottom of the plane moment of mentia auf sphere about center of mass I= 2 mR² initially ball have only potential energy which is conuerteg, when rolling into linear and kinetic energy apply energy conservation Pica = KE mgh = £mus + 2 1 ² V=RW I rolling in case of in case of alih mies +(ma? [ * Je mgh = Scanned Vite CamScanner 귀

☺ b) if h= 5.00m them wr ? wa Mage v = 19 x 9.8*5 = 8.37 m/s c) Total kimetic energy = rotational et translational . kimetic energy - In + muz nadmu? [ +1] To ke ( Via] mu? Rotational KE = tx & m p ve r = 1 m 2 let ay be the Rotational KIE. of Total K.E. x xoo - 16 mm) x y me = litet) $ x = x 100 = 28.57 28.57% of the sphere's total kinetic energy is gotational kinetic energy. Scanned with CamScanner

mg oo let us consider if fs be the friction force, which prouble rolling then allen apply equation of motion along x axis masimo- fs = more I de angular welocity Em = I am ã = R ă (im rolling) for pure rolling torque should be same and opposite in direction, fo R = Ima = RLO CS Scanned wito = RP CamScanner

mgsimo e a = mara a = mgsind . = constant m+ 7 h X = £at (from Meetion's lawo of motion ) for first complete revolution X= 2KR STR = 494? Q = lume for two complete gevalcetion x = 4(26R) eleto2 = code a te → tez les as so te - the plane - (2-1) SO - = I tact 1 Scanned with CamScanner 0141 -