Question

3. Consider a bicycle wheel of mass Mon which a small mass mis attached at the rim. The wheel considered to me a hoop with moment of inertia / -MR) is initially positioned on a horizontal surface with the small mass m at the point of contact, and the wheel can roll without slipping on the horizontal surface. Using as a generalized coordinate e, the angular position of the small mass m from the downward vertical, show that, for small oscillations (@< 1), the kinetic and potential energies of the system are T-5mx* f*ö+ MR di and V = mga Hence use Lagrange's equations of motion to show that satisfies MRÖ = -mg Re and hence that the angular frequency of small oscillations is Examine the form of w for (a) m <M and (b) m M and comment.

Answer 1

whell is - was earlin at rotated by an angle o, the bead which the bottom will move upward h = height of the bead after movemement oso 09 ) o OA OA = w 8o = OA = R6o8o OB Now h= R-R cose =R (1-oso) R 2 sine since o is small sino no :: 78 X 2x ? : R 42 - Now the potential energy = Img Red = v. Since the hoop is rotated so , rotational energy ! I tu? = 1 mR²w² = IMR²2 2 2 Tangential speed of the bead = RO & Energy = I met de

.... Now 20 Now Laf me?p'ö+ 4 Mato? – mgrig? Lagrangian egn. (only one variable t). - 667-ben 4 mečoʻx 26.0 + _MR?2016 heinamaan Now : () = { Me?ö + me! [v*ö + 620 6] {ma o'zip = MR?Ö t mR²o²ö + 2m R² g 6² - malo? angka = mRrë tme²o 0 + mR²e ² + mg Re Vanishes - MR?" tmg Ro =0 MRZÖ = -ma RO = 0 U un ön yangile 10-0 compare with this i twd = 0 - it was man ha wa vya