Concept explainers
A system consists of three identical 19.32-lb particles A, B, and C. The velocities of the particles are, respectively, vA = vA j, vB = vBi, and vC = vCk. Knowing that the angular momentum of the system about O expressed in ft · lb · s is HO = −1.2k, determine (a) the velocities of the particles, (b) the angular momentum of the system about its mass center G.
Fig. P14.11 and P14.12
(a)
Find the velocities of the particles.
Answer to Problem 14.11P
The velocity of particles A is
The velocity of particles B is
The velocity of particles C is
Explanation of Solution
Given information:
The angular momentum about point O is
Calculation:
The mass of three particles A, B, and C is equal.
Determine the weight of the identical particle.
Here, W is weight of each particle,
Substitute
Write the position vectors for the particles based on the given coordinate system:
Determine the angular momentum of the system about the origin using the Equation.
Here,
Substitute
Equating i, j, k components.
Find the velocity at point B as follows:
Thus, the velocity of particles B is
Find the velocity at point C as follows:
Substitute
Thus, the velocity of particles C is
Find the velocity at point A as follows:
Substitute
Thus, the velocity of particles A is
Determine position vector
Here,
Substitute
Find the position vector from the particles
Here,
Substitute
Find the position vector from the particles
Here,
Substitute
Find the position vector from the particles
Here,
Substitute
Express the linear momentum of particle A as follows:
Express the linear momentum of particle B as follows:
Express the linear momentum of particle C as follows:
(b)
Find the angular momentum
Answer to Problem 14.11P
The angular momentum
Explanation of Solution
Calculation:
Calculate the angular momentum about point G using the relation:
Here,
Substitute
Thus, the angular momentum
Want to see more full solutions like this?
Chapter 14 Solutions
VECTOR MECH. FOR EGR: STATS & DYNAM (LL
- A rocket weighs 2600 lb, including 2200 lb of fuel, which is consumed at the rate of 25 lb/s and ejected with a relative velocity of 13,000 ft/s. Knowing that the rocket is fired vertically from the ground, determine (a) its acceleration as it is fired, (b) its acceleration as the last particle of fuel is being consumed, (c) the altitude at which all the fuel has been consumed, (d) the velocity of the rocket at that time.arrow_forwardA pile-driver of mass 500 kg falls freely through a height of 1.5 m on to a pile of mass 200 kg. Determine the velocity with which the driver hits the pile. If, at impact, 3 kJ of energy are lost due to heat and sound, the remaining energy being possessed by the pile and driver as they are driven together into the ground a distance of 200 mm, determine (a) the common velocity immediately after impact, (b) the average resistance of the ground.arrow_forwardA 180-lb man and a 120-lb woman stand at opposite ends of a 300-lb boat, ready to dive, each with a 16-ft/s velocity relative to the boat. Determine the velocity of the boat after they have both dived, if (a) the woman dives first, (b) the man dives first.arrow_forward
- A billiard player sends the cue ball toward a group of three balls that are initially at rest and in contact with one another. After the cue ball strikes the group, the four balls scatter, each traveling in a different direction with different speeds as shown in the figure below. If each ball has the same mass, 0.16 kg, determine the total momentum of the system consisting of the four balls immediately after the collision. (Assume v1 = 0.33 m/s, θ1 = 70°, v2 = 0.49 m/s, θ2 = 30°, v3 = 0.21 m/s, v4 = 0.48 m/s.) magnitude ? (in kg m/s^2) direction ? (in degrees counterclockwise from the + x-axis)arrow_forwardConservation of Linear Momentum for a System of Particles Two spheres A and B (of mass m each) were fired vertically from the ground with different initial velocities at different times. When sphere A reached its summit, sphere B, with an upward velocity of 17 m/s, created a head on collision with A. The impact caused sphere A to break into two pieces, each of mass m/2. Knowing one piece reached Point C and the other piece reached point D after the collision, determine (a) the velocity of sphere B after the collision; (b) the angle and the speeds of the two pieces after the collision; and (c) the force exerted by sphere A on B if the collision occurred in 0.001 s.arrow_forwardA system consists of three identical 5-kg particles A, B, and C. Their position vectors (in meter) and velocity vectors (in m/s) are rA 0,0, 3 , v A v Aj , rB 2,2, 3 , vB v B i , and rC 1,4, 0 , vC v C k, respectively. Knowing that the angular momentum of the system about point O is HO 1.5k kg m 2s, determine (a) the velocities of the particles, (b) the angular momentum of the system about its center of mass G.arrow_forward
- A test machine that kicks soccer balls has a 5-lb simulated foot attached to the end of a 6-ft long pendulum arm of negligible mass. Knowing that the arm is released from the horizontal position and that the coefficient of restitution between the foot and the 1-lb ball is 0.8, determine the exit velocity of the ball (a) if the ball is stationary, (b) if the ball is struck when it is rolling towards the foot with a velocity of 10 ft/s.arrow_forwardA constant force P is applied to a piston and rod of total mass m to make them move in a cylinder filled with oil. As the piston moves, the oil is forced through orifices in the piston and exerts on the piston a force of magnitude kv in a direction opposite to the motion of the piston. Knowing that the piston starts from rest at t= 0 and x = 0, show that the equation relating x, v, and t, where x is the distance traveled by the piston and v is the speed of the piston, is linear in each of these variables.arrow_forwardDetermine the altitude reached by the spacecraft of Prob. 14.95 when all the fuel of its launching rocket has been consumed.Reference to Problem 14.95:A 540-kg spacecraft is mounted on top of a rocket with a mass of 19 Mg, including 17.8 Mg of fuel. Knowing that the fuel is consumed at a rate of 225 kg/s and ejected with a relative velocity of 3600 m/s, determine the maximum speed imparted to the spacecraft if the rocket is fired vertically from the ground.arrow_forward
- Principle of Angular Impulse and Momentum To apply the principle of angular impulse and momentum to find final speed and the time to reach a given speed. As shown, ball B, having a mass of 10.0 kg, is attached to the end of a rod whose mass can be neglected. Finding the final speed of the ball If the rod is 0.550 m long and subjected to a torque M=(1.95t2+3.75) N⋅m, where t is in seconds, determine the speed of the ball when t=4.80 s. The ball has a speed of v=2.25 m/s when t=0 Finding the time needed to reach a specific speed If the shaft is 0.250 m long, the ball has a speed of v=2.85 m/s when t=0, and the rod is subjected to a torque M=(3.40t+2.15) N⋅m, where t is in seconds, determine the time it will take for the ball to reach a speed of 5.80 m/s.arrow_forwardA system consists of three identical 14.32-lb particles A, B, and C. The velocities of the particles are, respectively, vA = vA j, vB = vBi, and vC = vCk. The angular momentum of the system about O expressed in ft·lb·s is HO = -1.2k. Determine the velocities of the particles. (You must provide an answer before moving to the next part.) The velocity of particle A is ( ft/s)j. The velocity of particle B is ( ft/s)i. The velocity of particle C is ( ft/s)k.arrow_forwardA system consists of three identical 12.32-lb particles A, B, and C. The velocities of the particles are, respectively, vA = vA j, vB = vBi, and vC = vCk. The angular momentum of the system about O expressed in ft·lb·s is HO = -1.2k. a) Determine the velocities of the particles. A, B, C?b) Determine the angular momentum of the system about its mass center G.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