Question

Problem 2 (10 pt.) A homogeneous sphere of mass m and radius b is rolling on an inclined plane with inclination angle ? in the gravitational field g. Follow the steps below to find the velocity V of the center of mass of the sphere as a function of time if the sphere is initially at rest. Bold font represents vectors. There exists a reaction force R at the point of contact between the sphere and the plane. The equations of motion of the sphere under an external force F and torque K (with respect to the center of mass of the sphere) are thus where r is the radius vector from the center of mass to the point of contact, P is the momentum of the sphere, M is its angular momentum, and r × R is the torque produced by the reaction force. These equations are familiar from earlier courses on mechanics and dynamics: the change of the momentum in time is a net force, the change of the angular momentum in time is a net torque. Take the axis perpendicular to the plane as the z axis, and the axis along which the sphere is moving as the y axis, whence r- -b2 (hat represents a unit vector). F-mg is the force of gravity acting vertically downward at the center of mass, whence K 0, we also have P = mV and M = iw (spherical top with moment of inertia 1 : mb2 with respect to its center of mass), where w is the angular velocity. The condition that the relative motion at the point of contact be zero (no sliding) at every instant is v = V +? × r-0 Differentiate the above condition with respect to time (r does not change), and substitute in it V and w derived from (1). Use a vector identity A x (B x C) (A C)B - (A B)C. Write the y and z components of the obtained equation (no motion along the r axis) and find the components Ry and Rx in terms of the components Fy and Fz. This will involve the components of a cross product. Then use the first equation in (1) to find the components Vy and V, which give the components of V by integration. If you want, you can check that differentiating with respect to time the conservation law for the total (kinetic plus potential) energy, where the kinetic energy is T-gmV2+ 1w2, and V and ? are related to one another by the condition of a pure rotation about an instantaneous axis, will also give the same V(t)

Answer 1

As per given data, Homogeneous sphere of mass m and radius B is rolling on an inclined plane with inclination angle alpha in the gravitational fields g. Velocity V = ? The given data is represented as below rightarrow By balancing the forces N = W cos alpha rightarrow 1 and ma = wsin alpha - mu N straightline 2 Torque is equal to 0.\r\n tau_0 = f(b) = mu(wcos alpha)b = I_0 alpha straightline3 I_0 = 2mb^2/5 straightline 4 For pure rolling a = b alpha straightline 5 By using equations 3, 4, and 5 b alpha = a = 5/2 log cos alpha here a represents acceleration of center of mass = dv/dt By interpretations we get, integral dv = integral(5/2 log cos alpha) dt V = (5/2 log cos alpha)t + c At t = 0, V = 0 therefore c = 0 Rightarrow V = (5/2 log cos alpha)t