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A homogeneous disk of radius R, depth H and mass...
Consider a cylinder of mass M, radius R and length L. (a) Calculate the inertia tensor...
Consider a cylinder of mass M, radius R and length L. (a) Calculate the inertia tensor for rotations about the center of mass in the frame where the z axis is along the axis of the cylinder. Use cylindrical coordinates, where x = r cos θ and y = r sin θ. (b) Find the inertia tensor in the frame where the center of the “bottom side” is at the origin with the z axis along the axis of the...
A homogeneous ring of radius R and mass m can roll on a horizontal surface without...
A homogeneous ring of radius R and mass m can roll on a horizontal surface without slipping. It is attached at the center to a spring of elastic constant k and rest length 1, and can oscillate on the horizontal plane. See the figure below for a schematic presentation. k (i) What is the number of degrees of freedom of the system? [2] (ii) Compute the moment of inertia of the ring about an axis perpendicular to it and going...
1. Consider a solid cone with uniform density p, height h, and circular base with radius R (hence mass M,sphR2). Let the vertex of the cone be the origin ofthe body frame. By symmetry, choose bas...
1. Consider a solid cone with uniform density p, height h, and circular base with radius R (hence mass M,sphR2). Let the vertex of the cone be the origin ofthe body frame. By symmetry, choose basis vectors e for the body frame such that the inertia tensor I, is diagonal. Will this rigid body with this body origin be an asymmetric top, a symmetric top, or a spherical top? Calculate the inertia tensor in this basis How will the inertia...
6. A disk with a mass M, a radius R, and a rotational inertia of I-...
6. A disk with a mass M, a radius R, and a rotational inertia of I- MR is attached to a horizontal spring which has a spring constant of as shown in the diagram. When the spring is stretched by a distance x and then released from rest, the disk rolls without slipping while the spring is attached to the frictionless axle within the center of the disk (a) Calculate the maximum translational velocity of the disk in terms of...
2. A homogeneous solid ball of mass m and radius r rolls without slip on a...
2. A homogeneous solid ball of mass m and radius r rolls without slip on a seesaw. The distance to their contact point from the hinge of the seesaw is x, and the angle of the seesaw w.r.t. the horizontal line is 8. The moment of inertia of the seesaw about its hinge is I, and the moment of inertia of the ball about its central axis is mr2. Find the kinetic energy T and the potential energy V of...
1 at 13:30. 2 Phys. 335 classical Mechanics Due: Jan. 10, 2020 (Homework II) Three equal...
1 at 13:30. 2 Phys. 335 classical Mechanics Due: Jan. 10, 2020 (Homework II) Three equal masses are located at coordinate points (1,0,0), (0, 1, 2) and (0,2,1). a) find the inertia tensor about the origin b) Find the principal moments of inert c) Find the principal axes.. 2. Calculate the principal moments of inertia and principal axes of a thin uniform circular disk of mass m and fadius a. 3. A homogeneous cylinder of mass in and radius r...
Free body diagram: 24 0.5r 0.5r No slip (a) An ec centric disk is rotating on the ground as shown in the figure above. The disk has radius r. The distance between the center of mass of th...
Free body diagram: 24 0.5r 0.5r No slip (a) An ec centric disk is rotating on the ground as shown in the figure above. The disk has radius r. The distance between the center of mass of the disk (denoted as C) to its geometric center (denoted as O) is 1 r. The angle of rotation of the disk is θ and the displacement at point O is x. The disk has mass m. The moment of inertia with respect...
Q7 (15 points): A solid cylinder of mass 5 kg and radius R 0.15 m rolls...
Q7 (15 points): A solid cylinder of mass 5 kg and radius R 0.15 m rolls without slipping on a horizontal surface and is accelerated to the right by a constant force F of magnitude 6 N that is applied at the cylinder by a massless rope as shown in the below figure. Find a) the magnitude of the acceleration of the center of mass of the cylinder, b) the magnitude of the angular acceleration of the cylinder about the...
Thanks so much ahead of time! A spherical shell of radius 0.43 m and mass 27...
Thanks so much ahead of time!
A spherical shell of radius 0.43 m and mass 27 kg initially rolls without slipping towards an incline at a linear speed of 9.1 m/s. If the ball rolls up an incline (inclined 35 degrees above the horizontal) without sliding. How high will it go? A professor holding a spinning wheel used for a demonstration brings the wheel to a stop by applying a constant, tangential friction force to the edge (his hand). The...
A solid uniform cylinder of mass M and radius R is at rest in the center...
A solid uniform cylinder of mass M and radius R is at rest in
the center on a flat slab of mass m and thickness d, which in turn
rests on a horizontal, frictionless table, as shown in the figure at
right below. A horizontal force of magnitude F is applied to the
slab, acting through the center of the slab, causing it to
accelerate to the right. The cylinder rolls without slipping on the
slab. (The direction of F...
A homogeneous ring of radius R and mass m can roll on a horizontal surface without slipping. It is attached at the center to a spring of elastic constant k and rest length 1, and can oscillate on the horizontal plane. See the figure below for a schematic presentation. k (i) What is the number of degrees of freedom of the system? [2] (ii) Compute the moment of inertia of the ring about an axis perpendicular to it and going...
1. Consider a solid cone with uniform density p, height h, and circular base with radius R (hence mass M,sphR2). Let the vertex of the cone be the origin ofthe body frame. By symmetry, choose basis vectors e for the body frame such that the inertia tensor I, is diagonal. Will this rigid body with this body origin be an asymmetric top, a symmetric top, or a spherical top? Calculate the inertia tensor in this basis How will the inertia...
6. A disk with a mass M, a radius R, and a rotational inertia of I- MR is attached to a horizontal spring which has a spring constant of as shown in the diagram. When the spring is stretched by a distance x and then released from rest, the disk rolls without slipping while the spring is attached to the frictionless axle within the center of the disk (a) Calculate the maximum translational velocity of the disk in terms of...
2. A homogeneous solid ball of mass m and radius r rolls without slip on a seesaw. The distance to their contact point from the hinge of the seesaw is x, and the angle of the seesaw w.r.t. the horizontal line is 8. The moment of inertia of the seesaw about its hinge is I, and the moment of inertia of the ball about its central axis is mr2. Find the kinetic energy T and the potential energy V of...
1 at 13:30. 2 Phys. 335 classical Mechanics Due: Jan. 10, 2020 (Homework II) Three equal masses are located at coordinate points (1,0,0), (0, 1, 2) and (0,2,1). a) find the inertia tensor about the origin b) Find the principal moments of inert c) Find the principal axes.. 2. Calculate the principal moments of inertia and principal axes of a thin uniform circular disk of mass m and fadius a. 3. A homogeneous cylinder of mass in and radius r...
Free body diagram: 24 0.5r 0.5r No slip (a) An ec centric disk is rotating on the ground as shown in the figure above. The disk has radius r. The distance between the center of mass of the disk (denoted as C) to its geometric center (denoted as O) is 1 r. The angle of rotation of the disk is θ and the displacement at point O is x. The disk has mass m. The moment of inertia with respect...
Q7 (15 points): A solid cylinder of mass 5 kg and radius R 0.15 m rolls without slipping on a horizontal surface and is accelerated to the right by a constant force F of magnitude 6 N that is applied at the cylinder by a massless rope as shown in the below figure. Find a) the magnitude of the acceleration of the center of mass of the cylinder, b) the magnitude of the angular acceleration of the cylinder about the...
Thanks so much ahead of time!
A spherical shell of radius 0.43 m and mass 27 kg initially rolls without slipping towards an incline at a linear speed of 9.1 m/s. If the ball rolls up an incline (inclined 35 degrees above the horizontal) without sliding. How high will it go? A professor holding a spinning wheel used for a demonstration brings the wheel to a stop by applying a constant, tangential friction force to the edge (his hand). The...
A solid uniform cylinder of mass M and radius R is at rest in
the center on a flat slab of mass m and thickness d, which in turn
rests on a horizontal, frictionless table, as shown in the figure at
right below. A horizontal force of magnitude F is applied to the
slab, acting through the center of the slab, causing it to
accelerate to the right. The cylinder rolls without slipping on the
slab. (The direction of F...
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