10-90 A homogeneous octant of a sphere is formed by ro- tating the quarter circle shown in Fig. P10-90 for 90° around the z-axis. Determine the moment of inertia of the body with respect to a y-axis through the mass center of the body. y
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- Two astronauts, each having a mass of 75.5 kg, are connected by a 10.0 m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 4.90 m/s. Treating the astronauts as particles, calculate each of the following. Center of gravity (a) the magnitude of the angular momentum of the system kg-m²/s (b) the rotational energy of the system KJ By pulling on the rope, the astronauts shorten the distance between them to 5.00 m. (c) What is the new angular momentum of the system? kg-m²/s (d) What are their new speeds? m/s (e) What is the new rotational energy of the system? KJ (f) How much work is done by the astronauts in shortening the rope? kJThe moment of inertia for a set of objects of mass m; rotating about a common axis is defined as 1 = Σm₁r? 2 where r, is the distance of the ith object to the axis of rotation. If there are many particles that make up a larger object then this sum transforms into an integral, 4-fff or a. I = dV, V where p is the mass density and V the volume of the object. In this exercise we will explore moment of inertia by rolling two objects down an incline plane in the Experimental Math Lab Space.A skateboard of mass m rests on the top of a hemispheric dome of radius R. After giving the skateboard a small kick, imparting an initial speed of vo, it begins rolling down the dome. Let the angle 0 indicate the location of the skateboard as it rolls down the surface of the dome. R 1. Eventually, the skateboard will lose contact with the dome. Given m, R, vo, and g, find the angle 0. at which the skateboard loses contact. Express your answer symbolically (i.e. without numbers). Check your work. Suppose you choose m = 1.5 kg, vo = g = 9.81 m/s. In this case, your answer should evaluate to 0. = 34.2°. 4.5 m/s, R = 4.3 m, and 2. Given m, R, and g, determine the range of speeds for which it is impossible to remain in contact with the wall at all.
- Among the elderly population, a sideway fall is a most frequent cause of hip fracture. An old man of mass 65 kg and height 1.7m was sent to Prince of Wales Hospital due to sideway fall. He slipped and fell laterally down with a straight body. Before falling, his centre of mass was 0.9 m above ground. His centre of mass was on the ground at the end of the fall. The radius of gyration about the anterior-posterior axis at his centre of mass was 0.55 m. (a) Calculate the initial potential energy before his fall. (b) Assuming his potential energy would be converted to linear kinetic energy of his centre of mass and rotational kinetic energy about his centre of mass, calculate the impact velocity v of his centre of mas right before he hit the ground. C. M. 1.7m 0.9m1 S X₁ X₂ E I TLL s F A person is pulling a rope and this results in a force ♬ that acts on their hand at point H, at an angle of 63 deg above the horizontal. Determine: The magnitude and direction of the moment about the shoulder (point S), in terms of F a) b) The magnitude and direction of the moment about the elbow (point E), in terms of F H V E EConsider a homogeneous rigid body with a non-uniform distribution of mass. The body is rotating about a fixed axis. Which of the following statements about the moments of inertia is correct? A) The principal axes of inertia will coincide with the axis of rotation. B) The moment of inertia about the axis of rotation will be the smallest possible. C) The moment of inertia about any axis parallel to the axis of rotation will be constant. D) The moment of inertia tensor will be diagonal, with all diagonal elements distinct. Choose the correct option and justify your choice.