Concept explainers
A projectile of mass m moves to the right with a speed νi (Fig. P11.51a). The projectile strikes and sticks to the end of a stationary rod of mass M, length d. pivoted about a frictionless axle perpendicular to the page through O (Fig. PH.51b). We wish to find the fractional change of kinetic energy in the system due to the collision, (a) What is the appropriate analysis model to describe the projectile and the rod? (b) What is the
(a)
The appropriate analysis model to describe the projectile and the rod.
Answer to Problem 11.51AP
The appropriate analysis model to describe the projectile and the rod is isolated system (angular momentum).
Explanation of Solution
Given info: The mass of the projectile is
From the conservation of the law of angular momentum the total angular momentum of the system remains constant when the net external torque acting on the system about the given axis is
It is clear from the given explanation and the figure that there is no external force that acts on the rod or the ball and so the angular momentum of the isolated system will remain conserved and so the model for the angular moment of the isolated system will be appropriate model for analysis of projectile and rod.
Conclusion:
Therefore, the appropriate analysis model to describe the projectile and the rod is isolated system (angular momentum).
(b)
The angular momentum of the system before an collision about an axis through
Answer to Problem 11.51AP
The angular momentum of the system before an collision about an axis through
Explanation of Solution
Given info: The mass of the projectile is
The distance from the origin to the ball is,
Here,
The formula to calculate the moment of inertia of the ball about the point
Here,
Substitute
The expression for the initial angular moment of the ball is,
Here,
The relation between the angular speed and the linear speed is,
Substitute
Substitute
Conclusion:
Therefore, the angular momentum of the system before an collision about an axis through
(c)
The momentum of inertia of the system about an axis through
Answer to Problem 11.51AP
The momentum of inertia of the system about an axis through
Explanation of Solution
Given info: The mass of the projectile is
From part (b) the moment of inertia of the ball is
The formula to calculate the moment of inertia of the rod about the point
The moment of inertia of the system when projectile sticks to the rod is,
Here,
Substitute
Conclusion:
Therefore, the momentum of inertia of the system about an axis through
(d)
The angular momentum of the system after collision.
Answer to Problem 11.51AP
The angular momentum of the system after collision is
Explanation of Solution
Given info: The mass of the projectile is
The expression for the final angular moment of the ball is,
Here,
From part (c) momentum of inertia of the system about an axis through
Substitute
Conclusion:
Therefore, the angular momentum of the system after collision is
(e)
The angular speed
Answer to Problem 11.51AP
The angular speed
Explanation of Solution
Given info: The mass of the projectile is
From the law of conservation of angular momentum the expression for the angular momentum is,
Substitute
Conclusion:
Therefore, the angular speed
(f)
The kinetic energy of the system before collision.
Answer to Problem 11.51AP
The kinetic energy of the system before collision is
Explanation of Solution
Given info: The mass of the projectile is
The formula to calculate the kinetic energy of the system is,
Here,
Substitute
Conclusion:
Therefore, the kinetic energy of the system before collision is
(g)
The kinetic energy of the system after collision.
Answer to Problem 11.51AP
The kinetic energy of the system after collision is
Explanation of Solution
Given info: The mass of the projectile is
The expression for the final kinetic energy of the system is,
Here,
From part (c) momentum of inertia of the system about an axis through
And from part (e) the angular speed
Substitute
Conclusion:
Therefore, the kinetic energy of the system after collision is
(h)
The fractional change of kinetic energy due to collision.
Answer to Problem 11.51AP
The fractional change in the kinetic energy of the system is
Explanation of Solution
Given info: The mass of the projectile is
The expression to calculate the change in the kinetic energy of the system is,
Here,
From part (g) kinetic energy of the system after collision is
Substitute
Thus, the change in the kinetic energy is
The formula to calculate the fractional change in the kinetic energy is,
Substitute
Conclusion:
Therefore, the fractional change in the kinetic energy of the system is
Want to see more full solutions like this?
Chapter 11 Solutions
EBK PHYSICS FOR SCIENTISTS AND ENGINEER
- A projectile of mass m moves to the right with a speed vi (Fig. P10.81a). The projectile strikes and sticks to the end of a stationary rod of mass M, length d, pivoted about a frictionless axle perpendicular to the page through O (Fig. P10.81b). We wish to find the fractional change of kinetic energy in the system due to the collision. (a) What is the appropriate analysis model to describe the projectile and the rod? (b) What is the angular momentum of the system before the collision about an axis through O? (c) What is the moment of inertia of the system about an axis through O after the projectile sticks to the rod? (d) If the angular speed of the system after the collision is , what is the angular momentum of the system after the collision? (e) Find the angular speed after the collision in terms of the given quantities. (f) What is the kinetic energy of the system before the collision? (g) What is the kinetic energy of the system after the collision? (h) Determine the fractional change of kinetic energy due to the collision. Figure P10.81arrow_forwardA wad of sticky clay with mass m and velocity vi is fired at a solid cylinder of mass M and radius R (Fig. P11.29). The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at a distance d R from the center. (a) Find the angular speed of the system just after the clay strikes and sticks to the surface of the cylinder. (b) Is the mechanical energy of the claycylinder system constant in this process? Explain your answer. (c) Is the momentum of the claycylinder system constant in this process? Explain your answer. Figure P11.29arrow_forwardA thin rod of length 2.65 m and mass 13.7 kg is rotated at anangular speed of 3.89 rad/s around an axis perpendicular to therod and through its center of mass. Find the magnitude of therods angular momentum.arrow_forward
- A wooden block of mass M resting on a frictionless, horizontal surface is attached to a rigid rod of length and of negligible mass (Fig. P11.27). The rod is pivoted at the other end. A bullet of mass m traveling parallel to the horizontal surface and perpendicular to the rod with speed v hits the block and becomes embedded in it. (a) What is the angular momentum of the bulletblock system about a vertical axis through the pivot? (b) What fraction of the original kinetic energy of the bullet is converted into internal energy in the system during the collision? Figure P11.27arrow_forwardA uniform rod of mass 200 g and length 100 cm is free to rotate in a horizontal plane around a fixed vertical axis through its center, perpendicular to its length. Two small beads, each of mass 20 g, are mounted in grooves along the rod. Initially, the two beads are held by catches on opposite sides of the rod’s center, 10 cm from the axis of rotation. With the beads in this position, the rod is rotating with an angular velocity of 10.0 rad/s. When the catches are released, the beads slide outward along the rod. (a) What is the rod’s angular velocity when the beads reach the ends of the rod? (b) What is the rod’s angular velocity if the beads fly off the rod?arrow_forwardA ring (2 kg, r = 1 m) rotates in a CW direction with initial angular velocity 20 s-1. A disk (4 kg, r = 1 m) rotates in a CCW direction with initial angular velocity 50 s-1. The ring and disk "collide" and eventually rotate together. Assume that positive angular momentum and angular velocity values correspond to rotation in the CCW direction.What is the initial angular momentum Li of the ring+disk system? kg m2 s-1 What is the final angular velocity ωf of the ring+disk system? s^-1arrow_forward
- Three light rods of negligible mass are joined to form an equilateral triangle of length L = 1.90 m. Three masses m, 6.00 kg, m, = 9.00 kg, and %D m3 3.00 kg are fixed to the vertices of this triangle as shown in the diagram below. Treat the masses as point particles. m, L L m, L/2 L/2 т, (a) What is the moment of inertia of the system about an axis lying in the plane of the triangle, passing through the midpoint of one side and the opposite vertex as shown by the dashed line? kg · m2 (b) What is the moment of inertia of the system about an axis passing through the center of the triangle and perpendicular to its plane? kg · m2arrow_forwardOn a frictionless table, a 0.56 kg glob of clay strikes a uniform 1.26 kg bar perpendicularly at a point 0.35 m from the center of the bar and sticks to it. If the bar is 1.06 m long and the clay is moving at 9.30 m/s before striking the bar, what is the final speed of the center of mass?arrow_forwardA rod of mass M = 3.25 kg and length L can rotate about a hinge at its left end and is initially at rest. A putty ball of mass m = 65 g, moving with speed v = 5.63 m/s, strikes the rod at angle θ = 55° from the normal at a distance D = 2/3 L, where L = 1.25 m, from the point of rotation and sticks to the rod after the collision. a) What is the initial angular momentum of the ball, in kilogram meters squared per second, right before the collision relative to the pivot point of the rod? b)What is the total moment of inertia If with respect to the hinge, of the rod-ball-system after the collision, in terms of the variables from the problem statement? c) What is the angular speed ωf of the system immediately after the collision, in radians per second?arrow_forward
- The figure shows a pendulum set up composed of mass A (100 g) and mass B (300 g). Mass A is pulled to the left until its height is 5 cm relative to its original position. After being released, it collides with mass B. Find the maximum angular displacement of mass B, α. Assume collision as central impact with coefficient of restitution of 0.75. State your answer in degrees.arrow_forward= 30°. Force F of magnitude = 45° and a velocity vector v of magnitude 3.8 m/s and angle 02 At the instant of the figure, a 1.6 kg particle P has a position vector r of magnitude 1.7 m and angle 01 1.6 N and angle 03 = 30° acts on P. All three vectors lie in the xy plane. (Express your answers in vector form.) 03 (a) What is the angular momentum of the particle about the origin? kg · m2/s (b) What is the torque acting on the particle about the origin? デ= N: m Additional Materialsarrow_forwardA ring (2 kg,r = 1 m) rotates in a CW direction with initial angular velocity 10 s-1. A disk (8 kg, r = 1 m) rotates in a CCW direction with initial angular velocity 50 s-1. The ring and disk "collide" and eventually rotate together. Assume that positive angular momentum and angular velocity values correspond to rotation in the CCW direction. What is the initial angular momentum L¡ of the ring+disk system? kg m2 s-1 Submit Answer Tries 0/3 What is the final angular velocity wf of the ring+disk system? s1 Submit Answer Tries 0/3arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningUniversity Physics Volume 1PhysicsISBN:9781938168277Author:William Moebs, Samuel J. Ling, Jeff SannyPublisher:OpenStax - Rice University