Derive an expression of conservation of moment in differential form.
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Derive an expression of conservation of moment in differential form.
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- The moment of a force about a point is equal to the sum of the moments of the components of the force about the point. O True O FalseA cylinder of mass M and radius 2R (careful!) is at rest ona rough table. A light string runs from the center of thecylinder in such a way as to allow the cylinder to bepulled horizontally. Said string runs over a disc of massm and radius R on a frictionless axle. The stringcontinues down over the disc and is connected to ahanging mass M.Once released from rest, the cylinder rolls withoutslipping on the table, and the string does not slip over the disc. What is the linear accelerationof the masses?HINT: You can NOT assume that Ffs = μfsFN here. The frictional force is just enough to preventslipping of the surfaces. You can NOT assume that the tensions are the same in each part of thestringR)A Solid homogeneous cylindor oF mass M and radiusr rolls without on a cart of mass m he cart is Connected by springs of Consfants Ki and K2 is free to slide ona Surface . a hori zontal Use Lagrange equabion to find the system's equation of motion.
- A rigid body consists of 12 identical thin rods of length a, forming the edges of a cube. Each of the rods has mass m and a uniform mass density. Calculate the moment of inertia tensor of the body. [Expect to use about half a page to answer the question.]A projectile of mass m moves to the right with a speed vi (see figure below). The projectile strikes and sticks to the end of a stationary rod of mass M, length d, pivoted about a frictionless axle perpendicular to the page through O. We wish to find the fractional change of kinetic energy in the system due to the collision. The moment of inertia of a rod is I = Md² and the moment of inertia of a particle is I = mr². 12 a. What is w, the angular speed of the system after the collision. mdvi Md²+1md² 12 b. What is the kinetic energy before the collision? Answer: KE¡ = m(v¡)². c. What is the kinetic energy after the collision? (0₁)² Answer: W= 1 m²d² Answer: KE₁ = = =+M₁2² +² md² 12 4 m O d M O 3Consider a rotating disk of radius R that has the exotic property that it can change its massper unit area as a function of r. Initially, its mass per unit area is given by ar3, where a is aconstant. What is the moment of inertia of this disk expressed in terms of its mass, M andradius R.
- A stick is resting on a concrete step with 211211 of its total length LL hanging over the edge. A single ladybug lands on the end of the stick hanging over the edge, and the stick begins to tip. A moment later, a second, identical ladybug lands on the other end of the stick, which results in the stick coming momentarily to rest at θ=36.1∘θ=36.1∘ with respect to the horizontal, as shown in the figure. If the mass of each bug is 3.263.26 times the mass of the stick and the stick is 10.3 cm10.3 cm long, what is the magnitude αα of the angular acceleration of the stick at the instant shown? Use g=9.81 m/s2.A particle of mass m is located at x = 1, y = 0,2 = 2. Find the tensor of inertia for the particle relative to the origin. The particle rotates about the z axis through a small angle a <<1 as shown below. Show that the moments of inertia are unchanged to second order in a but the products of inertia can change linearly with a.solve it on paper