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3 Consider a top with principle moments of inertia I3 > I2>. Initially it rotates with frequency wi about the axi...
Question 5 (20 marks) a) Develop a Lagrangian for the system shown, for small displacements from equilibrium when 6 (t)-0. The cylinder rotates on a fixed axis and has moment of inertia, J, about thi...
Question 5 (20 marks) a) Develop a Lagrangian for the system shown, for small displacements from equilibrium when 6 (t)-0. The cylinder rotates on a fixed axis and has moment of inertia, J, about this axis. [5 marks] x (t) k2 b) Then use Lagrange's Equation to determine the equations of motion. M(t) denotes an external moment applied to the cylinder. Also, express the equations in matrix form. [10 marks] c) Comment briefly on the dominant dynamic effects you would...
A cylinder with moment of inertia I1 rotates with angular velocity ω0 about a frictionless vertical axle.
A cylinder with moment of inertia I1 rotates with angular velocity ω0 about a frictionless vertical axle. A second cylinder, with moment of inertia I2, initially not rotating, drops onto the first cylinder (Fig. P8.55). Because the surfaces are
rough, the two eventually reach the same angular velocity,ω.Figure P8.55(a) Calculate ω. (Use I1
for I1, I2 for I2, and w0
for ω0 in yourequation.)(b) Show that energy is lost in this situation (Do this on
paper.Your instructor may ask you to turn in this work.), and...
A disk with moment of inertia 9.15 × 10−3 kg∙m^2 initially rotates about its center at...
A disk with moment of inertia 9.15 × 10−3 kg∙m^2 initially rotates about its center at angular velocity 5.32 rad/s. A non-rotating ring with moment of inertia 4.86 × 10−3 kg∙m^2 right above the disk’s center is suddenly dropped onto the disk. Finally, the two objects rotate at the same angular velocity ?? about the same axis. There is no external torque acting on the system during the collision. Please compute the system’s quantities below. 1. Initial angular momentum ??...
A spaceship is preparing to dock with a space station. Throughout this problem the space station...
A spaceship is preparing to dock with a space station.
Throughout this problem the space station is motionless in the
space frame (inertial frame). Prior to docking, some rotational
maneuvers are necessary for the spaceship. At t = 0, the angular
velocity of the spaceship is zero, and we have e_1 = x cap, e_2 = y
cap and e_3 = z cap. A number of small rocket engines are attached
to the exterior of the spaceship, and they can...
Heres example 10.2 (3) (30 points) In Example 10.2, the moment of inertia tensor for a...
Heres example 10.2
(3) (30 points) In Example 10.2, the moment of inertia tensor for a uniform solid cube of mass Mand side a is calculated for rotation about a corner of the cube. It also worked out the angular momentum of the cube when rotated about the x-axis - see Equation 10.51. (a) Find the total kinetic energy of the cube when rotated about the x-axis. (b) Example 10.4 finds the principal axes of this cube. It shows that...
Please work out 3 clearly and with units. Thanks! 1. A 10-g bullet travels vertically upwards...
Please work out 3 clearly and with units. Thanks!
1. A 10-g bullet travels vertically upwards at 1000 m/s, then strikes and passes through a 2.0-kg block initially at rest. The bullet emerges from the block with a speed of 400 m/s because it transfers momentum to the block. (a) How much momentum did the block gain? (b) How fast is the block now moving, and what is its kinetic energy? (c) Using energy considerations only, to what maximum height...
Question 5 (20 marks) a) Develop a Lagrangian for the system shown, for small displacements from equilibrium when 6 (t)-0. The cylinder rotates on a fixed axis and has moment of inertia, J, about this axis. [5 marks] x (t) k2 b) Then use Lagrange's Equation to determine the equations of motion. M(t) denotes an external moment applied to the cylinder. Also, express the equations in matrix form. [10 marks] c) Comment briefly on the dominant dynamic effects you would...
A cylinder with moment of inertia I1 rotates with angular velocity ω0 about a frictionless vertical axle. A second cylinder, with moment of inertia I2, initially not rotating, drops onto the first cylinder (Fig. P8.55). Because the surfaces are
rough, the two eventually reach the same angular velocity,ω.Figure P8.55(a) Calculate ω. (Use I1
for I1, I2 for I2, and w0
for ω0 in yourequation.)(b) Show that energy is lost in this situation (Do this on
paper.Your instructor may ask you to turn in this work.), and...
A spaceship is preparing to dock with a space station.
Throughout this problem the space station is motionless in the
space frame (inertial frame). Prior to docking, some rotational
maneuvers are necessary for the spaceship. At t = 0, the angular
velocity of the spaceship is zero, and we have e_1 = x cap, e_2 = y
cap and e_3 = z cap. A number of small rocket engines are attached
to the exterior of the spaceship, and they can...
Heres example 10.2
(3) (30 points) In Example 10.2, the moment of inertia tensor for a uniform solid cube of mass Mand side a is calculated for rotation about a corner of the cube. It also worked out the angular momentum of the cube when rotated about the x-axis - see Equation 10.51. (a) Find the total kinetic energy of the cube when rotated about the x-axis. (b) Example 10.4 finds the principal axes of this cube. It shows that...
Please work out 3 clearly and with units. Thanks!
1. A 10-g bullet travels vertically upwards at 1000 m/s, then strikes and passes through a 2.0-kg block initially at rest. The bullet emerges from the block with a speed of 400 m/s because it transfers momentum to the block. (a) How much momentum did the block gain? (b) How fast is the block now moving, and what is its kinetic energy? (c) Using energy considerations only, to what maximum height...
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