Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. D is the triangular region with vertices (0, 0), (2, 1), (0, 3); p(x, y) = 8(x + y) 48 m = (x, y) = 3 3 2 4

Mechanic Physics: Please make sure this is right In the figure, a ball of mass m = 56 g is shot with speed vi = 23 m/s (in the negative direction of an x axis) into the barrel of a spring gun of mass M = 263 ginitially at rest on a frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. Assume that the increase in thermal energy due to friction between the ball and the barrel is negligible. (a) What is the speed of the spring gun after the ball stops in the barrel? (b)What fraction of the initial kinetic energy of the ball is stored in the spring?

A water molecule consists of an oxygen atom and two hydrogen atoms. The two O—H bonds are each0.1 nm long and form an angle of 107◦ with each other. Where is the molecule’s centre of mass located?Consider the mass of the oxygen atom to be 16 times the mass of a hydrogen atom, and place yourhydrogen atoms along the x-axis of your coordinate system. [Hint: Draw your coordinate system!].

Problem 3: (a) Use spherical coordinates to find the center of mass (CM) of a uniform solid hemisphere of radius R, whose flat face lies in the ry plane with its center on the origin. [Note: dV = ² sin 0 dr do do.] (b) Use your result from part (a) to calculate the CM of a hemispherical "bowl" with outer radius R and inner radius kR, k < 1. (Depending on your work in part (a), you may not even need to do another integral.) (c) Use your result from the previous part to find the CM for an infinitely thin hemispherical shell of radius R.

Explorers in the jungle find an ancient monument in the shape of a large isosceles triangle as shown. The monument is made from tens of thou- sands of small stone blocks of density = 800 kg/m3 . The monument is 15.7 m high and 64.8 m wide at its base and is everywhere 3.60 m thick from front to back. Before the monument was built many years ago, all the stone blocks lay on the ground. (a) Choosing a coordinate system with x = 0 at the center of the pyramid, y = 0 at the base of the pyramid, and z = 0 at the face of the pyramid shown in the diagram, determine the coordinates of the center of mass of the pyramid in the x, y, and z directions. (b) How much work did laborers do on the blocks to put them in position while building the entire monument? Note: The gravitational potential energy of an object–Earth system is given by Ug = MgyCM, where M is the total mass of the object and yCM is the elevation of its center of mass above the chosen reference level.

The sphere of mass m1 = 5 kg falls from a height H = 1.4 m onto the homogeneous board of negligible mass. The board can rotate around a horizontal axis passing through point O, and a body of mass m2 = 3.9 kg is placed on it at a distance b = 1 m from the axis of rotation. The collision is perfectly inelastic. How high does the body of mass m2 rise after the collision (in m) if a = 0.5 m?

The height varies from h to zero according to this function: y(x) = h ( – 1)´ . The constants h and e replace 1.00 m and 3.00 m. There is also a thickness t and a density p. You need two integrals, the total mass and the center of mass. Possibly surprisingly, you don't actually need the numbers t, h, and p. Ax y(x) X The column at x has a mass Am = (density * volume) = y(x) p t Ax. You add all the Am values to get %3D the total mass M. The sum becomes an integral: М — pt y(x) dx For the center of mass, you add each column's x Am, and divide by M: pt Xc х у(x) dx Calculate xc. The only quantity you'll need is e = 5 m.

To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0 cm long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a 70.0 kg person, the mass of the upper leg is 8.60 kg, while that of the lower leg (including the foot) is 5.25 kg. Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) fully extended, and (b) bent at the knee to form a right angle with the upper leg.

The 0.5-kg ball is fired from the tube at A with a velocity of v = 6 m/s. If the coefficient of restitution between the ball and the surface is e = 0.8, determine the height h after it bounces off the surface. v = 6 m/s 2 m 130° B C h

The uniform 19-m pole has a mass of 150 kg and is supported by its smooth ends against the vertical walls and by the tension Tin the vertical cable. Compute the magnitudes of the reactions at A and B. B T 12 m 7 m A 14 m Answers: FA = N FB