Chapter 11, Problem 11.19P

A small bead of mass m=1g moves without friction along the inner surface of a deep spherical cup, which is motionless with respect to the Earth’s surface. The curvature of the cup has radius R=98 mm and center O. For ease of description, let’s pick a coordinate system with origin at the center of curvature of the cup, and a z axis pointing vertically down. a)The bead is deflected from equilibrium and released. Prove that the resulting motion is confined to a vertical plane (say the xz plane) containing the equilibrium and the initial position (point of release).(hint: use forces or torque & angular momentum) b)Prove explicitly that for small deviations from the equilibrium, the system in part “a” behaves as a simple harmonic oscillator (SHO) c)For the system in part “b”, derive the equation of motion x(t) based on the given initial conditions. Calculate the angular frequency, frequency, and period of these oscillations.

Set up an iterated integral moment of inertia about the y-axis of a triangular plate with vertices at (0,0), (0,4), and (2,2) if the density is given by a constant ρ = 5. (point mass, I = d^2m.) No need to evaluate the integral. Just set it up.

Find the center of mass, the moment of inertia about the coordinate axes, and the polar moment of inertia of a thin triangluar plate bounded by the lines y = x, y = - X, and y =2 if 8(x,y) = 5y+ 3. What are the coordinates for the center of mass? The center of mass is the point (Type an ordered pair. Type an integer or a simplified fraction.) What is the moment of inertia about the coordinate axes? y =(Type an integer or a simplified fraction.) y=(Type an integer or a simplified fraction.) What is the polar moment of inertia? o = (Type an integer or a simplified fraction.)

Problem 3 (a) Consider a homogeneous cylinder of mass M, radius R and height h. Show that the inertia tensor in the principal axis system has the form 4 (R² + ²) 0 0 Jab = 0 0 (R² +²) 0 0 MR² (b) Consider a right circular solid cone with radius r, height h and mass m. The density of the cone is constant: Z Calculate the centre of mass in the Cartesian coordinate system in which the base of the cone lies on the (x, y) plane centered at the origin (see figure). (c) Calculate the moment of inertia of the cone about its symmetry axis. (d) Find the principal axes of the cone. Is there any ambiguity in choosing the principal axis system?

Find the mass and the moment of the lamina with density δ ( x, y) = A, bounded by y = sin x, y = 0, x = 0 and x =π . A = 2x − 5. Sketch graph.

Time left 0:56:53 1. For the typical Oar (shown in the figure below) used in rowing boats, select an engineering material that would be lightweight and stiff. It must be strong enough to carry the bending moment exerted by the rower without breaking and it must have the right stiffness. The function is identified to be a beam, loaded in bending. The length L and stiffness S are fixed by the design, leaving the radius R as the only dimensional variable. The mass m of the oar is minimized by choosing materials with large values of the material index. E1/2 %3D

The direction of the moment around any point M, is defined by its moment axis, which is horizontal to the plane that contains the force F and its moment arm d.

Compute for the total moment of pt. A and pt. B Ps. with complete solution without any shortcuts, please

A force F of 36N acts on a box in the direction of the vector OP, where P (-5,7,–3) and O(0,0,0). Express the force as a vector. F = help (vectors) Find the angle between force F and the xy-plane. Answer in radians. help (angles)