Concept explainers
(a)
The appropriate analysis model to describe the projectile and the rod
(a)
Answer to Problem 51AP
The appropriate analysis model to describe the particle and the rod is a projectile motion and the rod behaves like an isolated system.
Explanation of Solution
Conclusion:
The mass is moving to the right with linear velocity, as it strikes to a stationary rod the mass will follow projectile motion with the rod. The other end of the rod will also rotate about the fixed point to balance the rod, so the
Thus, the appropriate analytical model to describe the particle and the rod is a projectile motion and the rod behaves like an isolated system.
(b)
The angular momentum of the system before the collision about an axis through
(b)
Answer to Problem 51AP
The angular momentum of the system before the collision about an axis through
Explanation of Solution
The angular momentum before the collision for a given system is the sum of angular momentum of ball and rod. But before the collision angular momentum of the rod is zero because the rod is stationary.
Consider the mass of the ball is concentrated at its centre so it behaves like a particle.
Write the expression for the angular momentum of the system before the collision as.
Substitute
Re-arrange the terms.
Here,
Write the expression for momentum for a particle as.
Here,
Conclusion:
Substitute
Thus, the angular momentum of the system before the collision about an axis through
(c)
The moment of inertia of the system about an axis through
(c)
Answer to Problem 51AP
The moment of inertia of the system about an axis through
Explanation of Solution
The moment of inertia for a given system is the sum of the moment of inertia for particle and rod.
Write the expression for moment of inertia for a particle about the fix point
Here,
Write the expression for moment of inertia for rod about the fix point
Here,
Write the expression for total moment of inertia for rod-particle system about the fix point
Here,
Substitute
Simplify the above equation for
Conclusion:
Thus, the moment of inertia of the system about an axis through
(d)
The angular momentum of the system after the collision.
(d)
Answer to Problem 51AP
The total angular momentum of the system after the collision is
Explanation of Solution
Write the expression for the angular momentum of the system after the collision.
Here,
Conclusion:
Substitute
Thus, the total angular momentum of the system after the collision is
(e)
The angular speed
(e)
Answer to Problem 51AP
The angular speed
Explanation of Solution
Write the expression for conserved angular momentum as.
Here,
Conclusion:
Substitute
Simplify the above expression for
Thus, the angular speed
(f)
The kinetic energy of the system before the collision.
(f)
Answer to Problem 51AP
The kinetic energy of the particle before the collision is
Explanation of Solution
The kinetic energy of the system before the collision is equal to kinetic energy of the particle because before the collision the rod is not in motion so the kinetic energy of the rod becomes zero. Therefore before the collision the kinetic energy of the system becomes the kinetic energy of the particle.
Write the expression for kinetic energy of the particle before the collision as.
Here,
Conclusion:
Thus, the kinetic energy of the particle before the collision is
(g)
The kinetic energy of the system after the collision
(g)
Answer to Problem 51AP
The kinetic energy of the system after the collision is
Explanation of Solution
Write the expression for rotational kinetic energy of the particle-rod system after the collision as.
Here,
Conclusion:
Substitute
Simplify the above expression as.
Thus, the kinetic energy of the system after the collision is
(h)
The fractional change of kinetic energy due to the collision.
(h)
Answer to Problem 51AP
The fractional change of kinetic energy due to the collision is
Explanation of Solution
Write the expression for change in kinetic energy for the system as.
Here,
Write the expression for fractional change in kinetic energy as.
Here,
Conclusion:
Substitute
Simplify the above expression for
Substitute
Thus, the fractional change of kinetic energy due to the collision is
Want to see more full solutions like this?
Chapter 11 Solutions
PHYSICS:F/SCI...W/MOD..-UPD(LL)W/ACCES
- 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 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_forwardOne end of a massless rigid rod of length is attached to a woodenblock of mass M restingon a frictionless, horizontal tabletop, and theother end is attached tothe table through apivot (Fig. P13.75). Abullet of mass m traveling with a speed v in adirection perpendicular to the rod and parallel to the table impacts the block and embeds itself inside. a. What is the angular momentum of this systemaround a vertical axis through the pivot after the collision?b. What is the fraction of the bullets initial kinetic energy thatis lost to internal energy duringthe collision? FIGURE P13.75arrow_forward
- Two particles of mass m1 = 2.00 kgand m2 = 5.00 kg are joined by a uniform massless rod of length = 2.00 m(Fig. P13.48). The system rotates in thexy plane about an axis through the midpoint of the rod in such a way that theparticles are moving with a speed of 3.00 m/s. What is the angular momentum of the system? FIGURE P13.48arrow_forwardA long, uniform rod of length L and mass M is pivoted about a frictionless, horizontal pin through one end. The rod is released from rest in a vertical position as shown in Figure P10.65. At the instant the rod is horizontal, find (a) its angular speed, (b) the magnitude of its angular acceleration, (c) the x and y components of the acceleration of its center of mass, and (d) the components of the reaction force at the pivot. Figure P10.65arrow_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 rigid, massless rod has three particles with equal masses attached to it as shown in Figure P11.37. The rod is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through the point P and is released from rest in the horizontal position at t = 0. Assuming m and d are known, find (a) the moment of inertia of the system of three particles about the pivot, (b) the torque acting on the system at t = 0, (c) the angular acceleration of the system at t = 0, (d) the linear acceleration of the particle labeled 3 at t = 0, (e) the maximum kinetic energy of the system, (f) the maximum angular speed reached by the rod, (g) the maximum angular momentum of the system, and (h) the maximum speed reached by the particle labeled 2. Figure P11.37arrow_forwardTwo astronauts (Fig. P10.67), each having a mass M, are connected by a rope of length d having negligible mass. They are isolated in space, orbiting their center of mass at speeds v. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum of the two-astronaut system and (b) the rotational energy of the system. By pulling on the rope, one of the astronauts shortens the distance between them to d/2. (c) What is the new angular momentum of the system? (d) What are the astronauts new speeds? (e) What is the new rotational energy of the system? (f) How much chemical potential energy in the body of the astronaut was converted to mechanical energy in the system when he shortened the rope? Figure P10.67 Problems 67 and 68.arrow_forwardA uniform solid sphere of mass m and radius r is releasedfrom rest and rolls without slipping on a semicircular ramp ofradius R r (Fig. P13.76). Ifthe initial position of the sphereis at an angle to the vertical,what is its speed at the bottomof the ramp? FIGURE P13.76arrow_forward
- A ball of mass M = 5.00 kg and radius r = 5.00 cm isattached to one end of a thin,cylindrical rod of length L = 15.0 cm and mass m = 0.600 kg.The ball and rod, initially at restin a vertical position and freeto rotate around the axis shownin Figure P13.70, are nudgedinto motion. a. What is therotational kinetic energy of thesystem when the ball and rodreach a horizontal position? b. What is the angular speed of the ball and rod when they reach a horizontal position? c. What is the linear speed of the centerof mass of the ball when the ball and rod reach a horizontalposition? d. What is the ratio of the speed found in part (c) tothe speed of a ball that falls freely through the same distance? FIGURE P13.70arrow_forwardA wheel of inner radius r1 = 15.0 cm and outer radius r2 = 35.0 cm shown in Figure P12.43 is free to rotate about the axle through the origin O. What is the magnitude of the net torque on the wheel due to the three forces shown? FIGURE P12.43arrow_forwardA square plate with sides 2.0 m in length can rotatearound an axle passingthrough its center of mass(CM) and perpendicular toits surface (Fig. P12.53). There are four forces acting on the plate at differentpoints. The rotational inertia of the plate is 24 kg m2. Use the values given in the figure to answer the following questions. a. Whatis the net torque acting onthe plate? b. What is theangular acceleration of the plate? FIGURE P12.53 Problems 53 and 54.arrow_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 EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
- Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning