Calculate the moment of inertia for a hollow sphere, whose density with homogeneous distribution is ρ (x, y, z) = k, where the larger radius is 3/4 larger than the smaller “a”.
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Calculate the moment of inertia for a hollow sphere, whose density with homogeneous distribution is ρ (x, y, z) = k, where the larger radius is 3/4 larger than the smaller “a”.
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- From a solid sphere of mass M and radius R, a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is MR² (b) 32 √2π (a) 4MR² 9√3π (c) MR² 16 √2 G4MR² (d) 3√3π politProblem 2: A tal, cylindrical chimney is toppling over as its base has ruptured. Let us say that the height H of the chimney is significantly larger than its thickness, so we may treat the chimney as a thin, rigid rod of length H. At the instant when the chimney is at angle 0o with respect to the vertical, what is the: (a) angular speed w of the top of the chimney? (b) radial component a, of the acceleration of the top? (c) tangential component aț of the acceleration of the top? (d) the specific value of 60 at which aț is equal to g?A circular cone with constant density 1, base radius 6, and height 8 is placed so the axis of symmetry is on the z-axis. A cylindrical hole of radius 1 is drilled through the axis of symmetry. Find the moment of inertia of the remaining shape. (The moment of inertia about the z-axis will be the same no matter where the shape is placed along the z- axis and whether the shape is pointing up or pointing down.)
- The wheels of a wagon can be approximated as the combination of a thin outer hoop of radius rh=0.262 m and mass 4.32 kg, and two thin crossed rods of mass 9.09 kg each. Imagine replacing the wagon wheels with uniform disks that are td=6.51 cm thick, made out of a material with a density of 7370 kg/m3. If the new wheel is to have the same moment of inertia about its center as the old wheel about its center, what should the radius of the disk be?Two uniform, solid spheres (one has a mass M and a radius R and the other has a mass M and a radius Rb=2R are connected by a thin, uniform rod of length L=2R and mass M. Note that the figure may not be to scale. Find an expression for the moment of inertia I about the axis through the center of the rod. Write the expression in terms of M, R, and a numerical factor in fraction form.A non-uniform disk of mass M and radius R has its mass distributed in such a way that the mass per unit area is a function of the radial distance r from the center of the disk: σ(r) = b r , where b is a constant to be determined. What is the rotational inertia of this disk about an axis through the center of mass and perpendicular to the plane of the disk? The area differential can be written as a ring of radius r and thickness dr: dA = 2πrdr.
- A wheel with a radius of 0.23 m and a mass of 7.31 kg spins about its axis at 6.53 radians per second. What is the kinetic energy of the wheel in J? Treat the wheel as a cylinder of constant density.A solid ball of radius r begins to roll down a hemisphere with radius R. Find the angular speed of the ball at the moment it leaves the surface of the hemisphere. Obviously, r < R.A child sits on a merry‑go‑round that has a diameter of 6.00 m. The child uses her legs to push the merry‑go‑round, making it go from rest to an angular speed of 18.0 rpm in a time of 37.0 s. What is the angular displacement Δθ of the merry‑go‑round, in units of radians (rad), during the time the child pushes the merry‑go‑round?
- A uniform thin spherical shell of mass M and radius R rotates about a vertical axis on frictionless bearings. A massless cord passes around the equator of the shell, over a pulley of rotational inertia I and radius r, and is attached to a small object of mass m. There is no friction on the pulley’s axle; the cord does not slip on the pulley. Determine the expression for the speed of the object after it falls a distance h from rest.Consider a bowling ball which is tossed down a bowling alley. For this problem, we will consider the bowling ball to be a uniform sphere of mass M and radius R, with a moment of inertia given by I = (2/5)MR2. The moment the ball hits the ground (t = 0), it is moving horizontally with initial linear speed v0, but not rotating (ω0 = 0). Due to kinetic friction between the ground and the ball, it begins to rotate as it slides. The coefficient of kinetic friction is µk. As the ball slides along the lane, its angular speed steadily increases. At some point (time tc), the “no-slip” condition kicks in, so that ω = v/R. After this, the ball moves with a constant linear and angular speed. Solve all parts of this problem symbolically. Use the rotational version of Newton’s second law to find an expression for the angular acceleration of the ball along the z-direction before the no-slip condition kicks in, αz. Your final expression should only involve the variables R, g, and µYou are trying to get a better feel for the effect of geometry and mass distribution on the moment of inertia. You have a solid disk and a thin ring, each of radius, r = 1.30 m, and mass, m = 73.0 kg. You mount both on fixed, horizontal frictionless axes about which they can spin freely. Then you spin them both. (a) How much work do you need to do to get each object to spin at 3.00 rad/s? (b) Let us assume that you have been causing them to spin by using a constant force applied tangentially to their circumferences. If the above speed is to be reached within 0.700 s, what is the magnitude of the force you need to apply to each object? (c) You next attempt to stop each object by pressing one finger on each side of each object, right at the outer edge. The coefficient of kinetic friction between each finger and the surface of each object is 0.300. Find the minimum force you have to apply to stop each object within 1.00 min.