Concept explainers
The kinetic energy lost of the plate when edge C of the plate hits the obstruction.
Answer to Problem 18.51P
The energy loss of circular disc after impact is
Explanation of Solution
Given information:
Mass of circular plate is
Expression of moment of inertia along the x-axis.
Here, the radius of the disc is.
Expression of moment of inertia along y-axis and z-axis.
Expression of moment of inertia along z-axis.
Expression for conservation of linear momentum.
Here,
Compare x-component of Equation (I) on both side.
Compare y-component of Equation (I) on both side.
Compare z-component of Equation (I) on both side.
Expression of relative position of C according to center of mass.
As
Expression of the velocity of circular disc C.
Substitute
Substitute
Compare y-component of Equation (V) on both sides.
Substitute
Expression of moment about center of mass.
Expression for kinetic energy of the circular plat before impact.
Expression for kinetic energy of the circular plat after impact.
Expression of energy loss.
Calculation:
Substitute,
Compare x-component of Equation (XII) on both side.
Compare y-component of Equation (XII) on both side.
Compare z-component of Equation (XII) on both side.
Substitute
Substitute
Equate Equation (XIII) and Equation (XIV).
Substitute
Substitute
Expression for velocity along x-axis, y-axis and z-axis.
Substitute 0 for
Substitute
Substitute
Substitute
Conclusion:
Thus, the energy loss of circular disc after impact is
Want to see more full solutions like this?
Chapter 18 Solutions
Vector Mechanics For Engineers
- A 3-kg bar AB is attached by a pin at D to a 4-kg square plate, which can rotate freely about a vertical axis. Knowing that the angular velocity of the plate is 120 rpm when the bar is vertical, determine (a ) the angular velocity of the plate after the bar has swung into a horizontal position and has come to rest against pin C, (b) the energy lost during the plastic impact at C.arrow_forwardShow that the angular momentum HB of a rigid body about point B can be obtained by adding to the angular momentum HA of that body about point A the vector product of the vector rA/B drawn from B to A and the linear momentum of the body: Further show that when a rigid body rotates about a fixed axis, its angular momentum is the same about any two points A and B located on the fixed axis (HA=HB) if, and only if, the mass center G of the body is located on the fixed axis.arrow_forwardA 1200-kg satellite designed to study the sun has an angular velocity of w0 = (0.050 rad/s)i + (0.075 rad/s)k when two small jets are activated at A and B in a direction parallel to the y axis. Knowing that the coordinate axes are principal centroidal axes, that the radii of gyration of the satellite are and that each jet produces a 50-N thrust, determine (a ) the required operating time of each jet if the angular velocity of the satellite is to be reduced to zero, (b ) the resulting change in the velocity of the mass center G.arrow_forward
- Denoting, respectively, by w, HO and T the angular velocity, the angular momentum, and the kinetic energy of a rigid body with a fixed point O, (a) ) prove that ) show that the angle 0 between w and HO will always be acute.arrow_forwardA square plate of side a and mass m supported by a ball-and-socket joint at A is rotating about the y axis with a constant angular velocity ω = ω 0 j when an obstruction is suddenly introduced at B in the xy plane. Assuming the impact at B to be perfectly plastic (e = 0), determine immediately after the impact (a ) the angular velocity of the plate, (b ) the velocity of its mass center G.arrow_forwardA uniform rod of mass m and length 5 a is bent into the shape shown and is suspended from a wire attached at point B. Knowing that the rod is hit at point A in the negative y direction and denoting the corresponding impulse by determine immediately after the impact (a) the velocity of the mass center G, (b) the angular velocity of the rod.arrow_forward
- A uniform 144-lb cube is attached to a uniform 136-lb circular shaft as shown, and a couple M with a constant magnitude is applied to the shaft when the system is at rest. Knowing that r = 4 in., L= 12 in., and the angular velocity of the system is 960 rpm after 4 s, determine the magnitude of the couple M.arrow_forwardThe double pulley shown has a mass of 15 kg and a centroidal radius of gyration of 160 mm. Cylinder A and block B are attached to cords that are wrapped on the pulleys as shown. The coefficient of kinetic friction between block and the surface is 0.2. Knowing that the system is at rest in the position shown when a constant force P = 200 N is applied to cylinder A , determine (a ) the velocity of cylinder A as it strikes the ground, (b ) the total distance that block B moves before coming to rest.arrow_forwardThe rotor of an electric motor has an angular velocity of 3570 rpm when the load and power are cut off. The 65-kg rotor, which has a centroidal radius of gyration of 175mm, until it reaches the maximum speed of 5250 rpm. Knowing that the kinetic friction results in a couple of magnitude 4.5 N.m exerted on the rotor, determine the number of revolutions that the rotor executes, before achieving its maximum speed and the time it took. (The final answer should be in two decimal places with correct units)arrow_forward
- A cylinder of radius r and weight W with an initial counterclockwise angular velocity w0 is placed in the corner formed by the floor and a vertical wall. Denoting by μk the coefficient of kinetic friction between the cylinder and the wall and the floor, derive an expression for the time required for the cylinder to come to rest.arrow_forwardThe rotor of an electric motor has an angular velocity of 3600 rpm when the load and power are cut off. The 120-lb rotor, which has a centroidal radius of gyration of 9 in., then coasts to rest. Knowing that kinetic friction results in a couple of magnitude 2.5 lb·ft exerted on the rotor, determine the number of revolutions that the rotor executes before coming to rest.arrow_forwardA homogeneous disk of weight W= 6 lb rotates at the constant rate W1= 16 rad/s with respect to arm ABC,which is welded to a shaft DCE rotating at the constant rate w 2= 8 rad/s. Determine the angular momentum HA of the disk about its center A.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY