Question

Problem 3 10 marks A particle of mass m slides down a frictionless incline of angle a, mass M and length L which is on a horizontal frictionless plane (see the Figure). If the particle starts initially from rest at the top of the incline, prove that the time for the particle to reach the bottom is given by, 2L(M + m sina) (M + m)g sina To setup the problem, choose a fixed vertical xy coordinate system as in the figure. Let R be the position vector of the center-of-mass C of the incline, A the constant) vector from C to the top of the incline, and the position vector of the particle relative to the top of the incline. Problem 4 10 marks Two solid spheres of total masses m, and my respectively collide such that at the instant of impact the x axis passes through the centers, and of the spheres. The collision is perfectly elastic (that is, (vi-v) = -(V-v). i). Find their velocities after impact (v and v) in terms of their velocities before impact (v and V2 or 04, V2.0, and ).

Answer 1

Stuton PrsbleM 3 Me System n ihline an bloik. Ne Rurzont entennad fora Cu The A enu Eystem zeno t System daes not mere hari zontalhy Lacom) O A T A-aceleration hhine (Sine the inine cnnet More vuhcally) Acceleraton k helatre ncline a (- axi-sino) bl^uk relatire to'meline acc eluration Sinuc Panalle ts e netine's su face Rbiai ab mA-aa) M (A) (aCom) = 0 Mm (mM) A ThN is Free bo dy dagrams 1

for inne 2 F MA FN - (2) FN Sin =MA 2 Fy FN-3) FN Mg for bleck Max -F sin - M (A - a x) - 4) masin -5) ma FN Sin = MА M +m MMa o FN = (MtM) sin Subsh t m (5) Mмаas'd -Mg= (M +M) sina Ma a a sin (M+M)Sin a 1 4

sh) + m six MI a (M+M)sino CM )g sino Mm siA2 Displa cemeut relative to fuline utat 2 As ing L 0 ((Mm)gsmdt 2 M+M Sin 2L (M+Ms (M+m)g sin ime tuen P=Pf onserring Momentu m M v, M,v,Ca,si k+m,' (a- sM)

p Sine Cllsion elautc G,-) - (-,). 2 , C) -c) Staee iumpaut in along anis AA momentim o eauh Aphere n Conlerve) along y-ans M, , sn - (e) m22Sh ) wVSiMe 2 2 Canserraton f momeutirm along am m,v,' Co o() from ('a -v'cer+(a, - V Core Snbst tle m (4) myCoe,ar) Vi

Similarky fm ) Substile m (4) 2mi Cose(m-M) FhAl answes are. (m-M) Cos, 2m Cot Sin 14 m 2M,C wm,-M) V ( 2- MIm