Question

A sphere of radius ? is constrained to roll without slipping on the lower half of the inner surface of a hollow cylinder of inside radius R. Determine the Langrangian function, the equation of constraints, and Lagrange’s equation of motion. Find the frequency of small oscillations.

please explain what/why you're doing what you're doing. thank you

Answer 1

brion CR-18=10 Noslip condition CR ((RS) 0 7 sh-R/R-seso h = R(k 2 generalized coordinaty (0,0) but constraint reduces this to one Coordinate o. There are two co-ordinates are 0,6 but due to dependency between them (no slip) then this reduces the degree of freedom by one, and there is one generalized coordinate o The constraint of nostip means f(0,ø) = (R-8) 0-30=0 which means the center of the small disk more in speed the same as the point of the disk that moves on the edge of the larger cylinder as shown in the figaban Ta tId² + m (R-839 Uzmgh = mg [R-1R-3) (so] using I = 25 mg , 8= Rusi 2 To dla me?) RP) ] +(8 m(RS - ŚM (R-8) 352 +3m CR-D]262 T=7om CR-8282 ?

Hence Lapo - Dom(R-2 52 mg ER-R-37 eso] de = m (R-8320 Therefore the equation of motion sont les como tmg CR-) smoco öt sgozo o at TCR-8) 158 wa xangulay oscillation JTCR-) The freguery of . we know that anfaw 58 f = lo ha = 22 V TCR-5) - 1.sg f = FCR-5)