Question

A 2000-N crate is to be held in place on a ramp that rises at 30.0° above the horizontal (see figure). The massless rope attached to the crate makes a 22.0° angle above the surface of the ramp. The coefficients of friction between the crate and the surface of the ramp are uk 0.450 and us 0.650. The pulley has no appreciable mass or friction. = LLLLLL w = ? 22.09 Crate Ramp 30.0°

a.) What is the MAXIMUM weight w that can be used to hold this crate stationary on the ramp? Include free-body diagrams of the two masses.

b.) Now assume that the weight w=3000N. What is the acceleration of the crate?

c.) Find the speed of the masses after the weight w=3000N has dropped a distance of 1.6m after being released from rest. Find this speed two ways: using kinetmatics and the acceleration derived in the previous subquestion, and using energy considerations.

Answer 1

at (lanza tak din22°) -ing dozu - Yxng to be ama ( The fod of Weight T T= am' La & w, m' w v ركه 3 T = o-ma mis from 11 and noor w g 3000 9-8 = 306.12 w-n's) (Cs at Situbonza) om. Conan دلار (بر اور nik - mg n20 - Sping costs zmet ad aw زرند زنده اور اس کو 45.2009 6 almt im' (las 23 + Gefrizzo, .] a tol Cos zze meg tanany -ngfrau ant jo tu ex ng lesson a 3000 Cosazot 0.95 Inze )-2000 204.087 308.12 (Cas220 "45486220) دیو و و و و و a=2-795mit а 41 1507. ac 539.51

w = y2+zas. e2=0+2x2-795*1.6 Benthalt Speed 43 le= 3.991 mst unus

@ FBD of crate A quilibrium 'In case of max weight the frictech force fazzo ts Those ing lista) ts=0 ng te=Ysl dwell act downward the incline MAT Enzza ng Cos 30o =0 from T M= mgles 300 T fazzo we know fs = Yen= Is ng G30 - UST fazzo putting this in eparation o Teos2ro-ing 300+ Het fingsso In 30 - Ysing Cos 30 & T (1322°-Ms finas) = ingin3oot Is ng lossi Ta mg &n30°t yo ng Coszu? Cosso - Yoth 2000 0 + 0.65 * 2000 ft 220 12 0.9272-0.24 35

. Corresponding to weight max T=w T=3109 BIN The fod of weight w=3109.314 •T aso (at equilibrium) h) map. weight Tw=3109-3104 ② Suppose tome how the motion is just Harted, the kinetic freeferon force will act on the crate, there will be acceleration Cupward the ensline) even for the weight = 3000 M (< Wm=3109.5%) The FBD of the Crate ma 2010 g 204.og Tasaro-mg finger fx=ma M +Tfnorº-mg Cas 30º = 0 to ung di= Sen log M2 mg Cas 30°- Ton 220 Therefore T63 22° - mg dnao - Yk (ong Cess"-t snosila