Find the moments of inertia (J1,J2) of a homogeneous ring of circular shape with respect to both axes. The mass and radius (m,R) of the ring are known
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Find the moments of inertia (J1,J2) of a homogeneous ring of circular shape with respect to both axes. The mass and radius (m,R) of the ring are known
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- Scenario: A 0.31 kg meter stick hangs by a hole drilled at the 0.31 meter mark (the center of the stick is at the 0.50 meter mark, of course). The stick is swung up to an angle of 74 degrees from vertical. For all parts, be sure to double-check your distances! (a) What is the torque on the meter stick due to gravity? N m (b) What is the moment of inertial of the stick about the pivot? (parallel axis theorem) kg m? (c) What is the angular acceleration of the stick? rad/s? (d) What is the linear acceleration of the end of the stick? m/s²An asteroid can be modeled as a right circular cylinder of radius 150 m and length 500 m with uniform density of 2200kg/m^3. It is rotating about an axis perpendicular to its length through its center of mass at one rotation per 5 minutes.If a space probe lands at one end and begins to fire its engines in the direction of rotation to slow it down(A) What force is it exerting at the end of the asteroid if it ejects 0.15 kg/s at a speed of 800 m/s?(B) How long will it take to stop the asteroid’s rotation?A 117 kg horizontal platform is a uniform disk of radius 1.67 m and can rotate about the vertical axis through its center. A 62.7 kg person stands on the platform at a distance of 1.11 m from the center, and a 29.5 kg dog sits on the platform near the person 1.39 m from the center. Find the moment of inertia of this system, consisting of the platform and its population, with respect to the axis. moment of inertia: kg · m?
- 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 139-kg platform-oriented horizontally consists of a uniform disk of radius 1.85 m and can rotate about the vertical axis through its center. A 63.3-kg person stands on the platform at a distance of 1.17 m from the center and a 27.5-kg dog sits on the platform near the person, 1.39 m from the center. Find the moment of inertia of this system, consisting of the platform and its population, with respect to the axis.A 109 kg horizontal platform is a uniform disk of radius 1.57 m and can rotate about the vertical axis through its center. A 62.9 kg person stands on the platform at a distance of 1.15 m from the center, and a 28.7 kg dog sits on the platform near the person 1.39 m from the center. Find the moment of inertia of this system, consisting of the platform and its population, with respect to the axis. moment of inertia: kg · m²
- A space station is constructed in the shape of a hollow ring of mass 4.35 x 104 kg. Members of the crew walk on a deck formed by the inner surface of the outer cylindrical wall of the ring, with radius 110 m. At rest when constructed, the ring is set rotating about its axis so that the people inside experience an effective free-fall acceleration equal to g. (See figure below.) The rotation is achieved by firing two small rockets attached tangentially to opposite points on the rim of the ring. (a) What angular momentum does the space station acquire? |kg · m2/s (b) For what time interval must the rockets be fired if each exerts a thrust of 140 N?A constant force F of the same magnitude is applied a solid sphere, a solid disk, and a solid rod, as shown in the figure. The sphere, the cylinder and the rod have the same mass M and are initially at rest. The force is always applied normal to the vector connecting the center of mass of the object and the point of force application. Each body is constrained to rotate about a line through its center as shown in the figure. Which of the bodies will have the smallest angular acceleration? Neglect the friction. F F rod а. b. cylinder sphere С.