A uniform slender rod of length L is dropped onto rigid supports at A and B. Because support B is slightly lower than support A, the rod strikes A with a velocity
Fig. P17.111
(a)
Find the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support A.
Answer to Problem 17.111P
The angular velocity of the rod when mass center immediately after the rod strikes support A is
The velocity of the rod when mass center immediately after the rod strikes support A
Explanation of Solution
Given information:
The length of uniform slender rod is L.
The mass of uniform slender rod is m.
The initial velocity of bar before it strikes B is
Calculation:
Write the equation of centroidal moment of inertia
The impact is perfectly elastic at both A and B. Therefore, the coefficient of restitution is one
Write the impact condition for the given system after the rod strikes support A.
Here,
Write the equation of velocity of rod
Here,
Substitute
Consider the impulse and momentum principle.
Sketch the impulse and momentum diagram for the first impact at A of the bar as shown in Figure (1).
Here,
Refer Figure (1).
Take moment about A (positive sign in clockwise direction).
Substitute
Simplify the Equation:
Thus, the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support A is
Find the velocity of the rod when mass center immediately after the rod strikes support A using Equation (1).
Substitute
Thus, the velocity of the rod when mass center immediately after the rod strikes support A
(b)
Find the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support B.
Answer to Problem 17.111P
The angular velocity of the rod when mass center immediately after the rod strikes support B is
The velocity of the rod when mass center immediately after the rod strikes support B is
Explanation of Solution
Calculation:
Write the impact condition for the given system after the rod strikes support B.
Here,
Write the equation of velocity of rod
Here,
Substitute
Consider the impulse and momentum principle.
Sketch the impulse and momentum diagram for impact at B of the bar as shown in Figure (2).
Here,
Refer Figure (2).
Take moment about B (positive sign in clockwise direction).
Substitute
Thus, the angular velocity of the rod when mass center immediately after the rod strikes support B is
Find the velocity of the rod when mass center immediately after the rod strikes support B using Equation (2).
Substitute
Thus, the velocity of the rod when mass center immediately after the rod strikes support B
(c)
Find the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support A again.
Answer to Problem 17.111P
The angular velocity of the rod when mass center immediately after the rod strikes support A again is
The velocity of the rod when mass center immediately after the rod strikes support A again is
Explanation of Solution
Calculation:
Write the impact condition for the given system after the rod again strikes support A.
Here,
Write the equation of velocity of rod
Here,
Substitute
Consider the impulse and momentum principle.
Sketch the impulse and momentum diagram for second impact at A of the bar as shown in Figure (3).
Refer Figure (3).
Take moment about A (positive sign in clockwise direction).
Substitute
Thus, the angular velocity of the uniform slender rod after rod again strikes support A support B is
Find the velocity of rod at its mass center after the rod again strikes support A using Equation (3).
Substitute 0 for
Thus, the velocity of rod at its mass center after the rod again strikes support A is
Want to see more full solutions like this?
Chapter 17 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
Additional Engineering Textbook Solutions
Thinking Like an Engineer: An Active Learning Approach (3rd Edition)
HEAT+MASS TRANSFER:FUND.+APPL.
Fox and McDonald's Introduction to Fluid Mechanics
Introduction To Finite Element Analysis And Design
Mechanics of Materials
Manufacturing Engineering & Technology
- A 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_forwardA 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_forwardSphere A Mass m and radius r rolls without slipping with a velocity V1 on a horizontal surface when it hits squarely an identical sphere B that is at rest. Denoting by μk the coefficient of kinetic friction between the spheres and the surface, neglecting friction between the spheres, and assuming perfectly elastic impact, determine (a ) the linear and angular velocities of each sphere immediately after the impact, (b) the velocity of each sphere after it has started rolling uniformly.arrow_forward
- A 18-kg rear hatch of a vehicle opens as shown in the figure and can be modeled as a uniform 0.6-m long slender rod. Knowing that the tailgate is released from rest in the position shown in the figure, determine the angular velocity of the tailgate as it impacts the car body.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_forwardEach of the gears A and B has a mass of 675 g and a radius of gyration of 40 mm, while gear C has a mass of 3.6 kg and a radius of gyration of 100 mm. Assume that kinetic friction in the bearings of gears A, B C produces couples of constant magnitude 0.15 N.m, 0.15 N.m, 0.3 N.m, respectively. Knowing that the initial angular velocity of gear C is 2000 rpm, determine the time required for the system to come to rest.arrow_forward
- The 100-kg projectile shown has a radius of gyration of 100 mm about its axis of symmetry Gx and a radius of gyration of 250 mm about the transverse axis Gy. Its angular velocity v can be resolved into two components; one component, directed along Gx, measures the rate of spin of the projectile, while the other component, directed along GD, measures its rate of precession. Knowing that θ= 6° and that the angular momentum of the projectile about its mass center G is determine (a) the rate of spin, (b) ) the rate of precession.arrow_forwardTwo uniform rods, each of mass m, form the L-shaped rigid body ABC, which is initially at rest on the frictionless horizontal surface when hook D of the carriage E engages a small pin at C. Knowing that the carriage is pulled to the right with a constant velocity v0, determine immediately after the impact (a) the angular velocity of the body, (b) the velocity of corner B. Assume that the velocity of the carriage is unchanged and that the impact is perfectly plastic.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