29 May Lab 10 – BALLISTIC PENDULUMTHEORY:A projectile launcher can rep
Lab 10 – BALLISTIC PENDULUMTHEORY:A projectile launcher can repeatedly launch an object with a constant initial speed. The object can then collide with, and become attached to, a ballistic pendulum undergoing a completely inelastic collision. In this type of collision, linear momentum is conserved, but total mechanical energy is not conserved. After the collision, as the pendulum arm swings upwards, the total mechanical energy of the system is also conserved if neglecting resistive forces. Consider the initial state of the system where the object (small sphere) with mass m is launched at a large mass M connect to one or more strings of length L as in the figure below: Figure 1. Sphere launched towards Ballistic Pendulum Block at rest. The mass M is initially at rest, and the mass m has an initial velocity that is unknown. An inelastic collision then occurs between the sphere and the block and the sphere becomes imbedded in the block. Immediately after the collision, the block and sphere have a velocity that is determined by momentum conservation. See figure 2. ‘ Figure 2. Sphere and Block immediately after collision. The Ballistic Pendulum will then swing up conserving Mechanical Energy and converting its kinetic energy into potential energy. The Ballistic Pendulum will reach a maximum height h before swinging down again. See Figure 3. Figure 3. Ballistic Pendulum at its maximum swing position.Initial Calculations: Given a maximum height h attained by the Ballistic Pendulum swing, use Conservation of Energy to derive an expression for the velocity of the Ballistic Pendulum immediately after the collision.Determine this velocity for the case where . Given a velocity for the Ballistic Pendulum immediately after the collision, use Conservation of Momentum to derive an expression for the initial velocity of the sphere before the collision. Using the numerical values for Question 1 above, determine the initial velocity of the sphere for this case. This is an inelastic collision and in an inelastic collision kinetic energy is not conserved. Determine the difference in the kinetic energy of the system immediately before the collision to that immediately after the collision. What happens to this energy? Lab Activity:Open the following link for a simulation of the Ballistic Pendulum: https://ophysics.com/e3.htmlSet the initial parameters for the simulation ofMeasure the maximum height reached by the Ballistic Pendulum. Compare this with what you would predict from theory. Show calculation of the percent error PE value the height reached. For a initial mass of the sphere and Ballistic Pendulum of ; what initial velocity of the sphere results in a maximum height of ? Show a calculation for your prediction, and test your result using the Simulation. Increasing the mass of the Ballistic Pendulum to ; what initial velocity now results in the same maximum height for the Ballistic Pendulum, ? Show your calculation and test your prediction. Take the Ballistic Pendulum Quiz at the following link: https://ophysics.com/e4.htmlSet the mass of the sphere and Ballistic Pendulum to Adjust the initial velocity bar until the Ballistic Pendulum reaches a maximum height of Compute the initial velocity value and enter it into the box and see if you are correct. Reflection on the resultThe Ballistic Pendulum makes use of an inelastic collision between the sphere and the Ballistic Pendulum. If it were to be an elastic collision between the sphere and the Pendulum block, would the Ballistic Pendulum rise higher or lower than for the inelastic collision? Show a calculation to justify your answer using, Conclusion:Lab 10 – BALLISTIC PENDULUMTHEORY:A projectile launcher can repeatedly launch an object with a constant initial speed. The object can then collide with, and become attached to, a ballistic pendulum undergoing a completely inelastic collision. In this type of collision, linear momentum is conserved, but total mechanical energy is not conserved. After the collision, as the pendulum arm swings upwards, the total mechanical energy of the system is also conserved if neglecting resistive forces. Consider the initial state of the system where the object (small sphere) with mass m is launched at a large mass M connect to one or more strings of length L as in the figure below: Figure 1. Sphere launched towards Ballistic Pendulum Block at rest. The mass M is initially at rest, and the mass m has an initial velocity that is unknown. An inelastic collision then occurs between the sphere and the block and the sphere becomes imbedded in the block. Immediately after the collision, the block and sphere have a velocity that is determined by momentum conservation. See figure 2. ‘ Figure 2. Sphere and Block immediately after collision. The Ballistic Pendulum will then swing up conserving Mechanical Energy and converting its kinetic energy into potential energy. The Ballistic Pendulum will reach a maximum height h before swinging down again. See Figure 3. Figure 3. Ballistic Pendulum at its maximum swing position.Initial Calculations: Given a maximum height h attained by the Ballistic Pendulum swing, use Conservation of Energy to derive an expression for the velocity of the Ballistic Pendulum immediately after the collision.Determine this velocity for the case where . Given a velocity for the Ballistic Pendulum immediately after the collision, use Conservation of Momentum to derive an expression for the initial velocity of the sphere before the collision. Using the numerical values for Question 1 above, determine the initial velocity of the sphere for this case. This is an inelastic collision and in an inelastic collision kinetic energy is not conserved. Determine the difference in the kinetic energy of the system immediately before the collision to that immediately after the collision. What happens to this energy? Lab Activity:Open the following link for a simulation of the Ballistic Pendulum: https://ophysics.com/e3.htmlSet the initial parameters for the simulation ofMeasure the maximum height reached by the Ballistic Pendulum. Compare this with what you would predict from theory. Show calculation of the percent error PE value the height reached. For a initial mass of the sphere and Ballistic Pendulum of ; what initial velocity of the sphere results in a maximum height of ? Show a calculation for your prediction, and test your result using the Simulation. Increasing the mass of the Ballistic Pendulum to ; what initial velocity now results in the same maximum height for the Ballistic Pendulum, ? Show your calculation and test your prediction. Take the Ballistic Pendulum Quiz at the following link: https://ophysics.com/e4.htmlSet the mass of the sphere and Ballistic Pendulum to Adjust the initial velocity bar until the Ballistic Pendulum reaches a maximum height of Compute the initial velocity value and enter it into the box and see if you are correct. Reflection on the resultThe Ballistic Pendulum makes use of an inelastic collision between the sphere and the Ballistic Pendulum. If it were to be an elastic collision between the sphere and the Pendulum block, would the Ballistic Pendulum rise higher or lower than for the inelastic collision? Show a calculation to justify your answer using, Lab 11 – Torque and Rotational InertiaTHEORY: Torque is the rotational analogue to force and is responsible for why an object’s rotational motion changes. The definition of torque is The magnitude of the torque is given by where is the angle between the force direction and the vector that points from the axis of rotation to the point where the force is applied. For a non-zero Torque you need: . A rigid body that has a non-zero net torque applied to it will experience a non-zero angular acceleration according to Newton’s 2nd Law of Rotational Motion, given by where I is the Rotational Inertia or Moment of Inertia of the rigid body; the rotational analogue of mass for translational motion and is the angular acceleration. The angular acceleration results in a change in the angular velocity of the object. If the torque is constant, then the angular acceleration is constant and the angular velocity changes with time according to and the angular displacement changes according to The rotational inertia of an object depends on its mass and how the mass is distributed about a defined rotational axis. For certain geometric objects the rotational inertia about an axis through the center of mass can be determine by integration. For a hoop or thin-walled cylindrical shell of mass M and radius R, the rotational inertia about an axis through its CM and normal to its face is given by For a solid disk or solid cylinder of mass M and radius R , the rotational inertia about an axis through the CM and normal to its face is given byFor a solid sphere of mass M and radius R , the rotational inertia about an axis through its CM is given by For example, a mass m is suspended by a string wrapped around a solid disk or solid cylinder as in Figure 1, would exert a torque about an axis through the disk’s center of mass causing the disk to rotate. Figure 1. A mass m suspended from a solid cylinder (disk) of mass M and radius R exerts a torque. Rotational kinetic energy is also associated with rotating objects is given by which is the rotational analogue of .If more than one torque act on a rigid body, the net torque is vector sum of the torques. If all the forces and radial distances for the axis of rotation are in a single plane, then torques can be given positive or negative values depending on whether they would result in CCW rotations (+) or CW rotations (-). In Figure 2, two masses are suspended from a solid disk connected by a string wrapped around the outside of the disk. In this case, the tensions are different on the different sides, resulting in a non-zero net torque about rotational axis and an angular acceleration for the disk. Figure 2. Two masses suspended from a solid cylinderInitial Calculations: For the setup shown in figure 1, the suspended mass is and the pulley is a thin-walled cylindrical shell of mass and radius . Set up a free body diagram for the suspended mass labelling the forces acting on the mass m and using Newton’s second law of linear motion, write and expression for the acceleration of the mass in terms of the tension in the string and the mass. For the same setup, using Newton’s 2nd Law for rotational motion, write down an expression for the angular acceleration of the pulley in terms of its mass and radius and the tension in the string. Assuming the tension in part (a) is the same as in part (b), determine the angular acceleration of the pulley, the linear acceleration of the suspended mass and the string that is connected to it, and the tension in the string. Determine the magnitude of the torque acting on the pulley in problem above. Consider the setup in Figure 2, with the left suspended mass is the right suspended mass is , and the pulley is a thin-walled cylindrical shell of mass and radius . Set up free-body diagrams for each suspended mass labelling the forces acting on the mass. For each free-body diagram, use Newton’s second law of linear motion to write an expression for the acceleration of the mass in terms of the tension in the string on that side of the pulley and the mass. For the same setup, using Newton’s 2nd Law for rotational motion, write down an expression for the angular acceleration of the pulley in terms of its mass and radius and the tension in the string. Simultaneously solving the two equations in part (e) and the equation in part (f), determine the angular acceleration of the pulley, the linear acceleration of the suspended masses and the string that is connected to it, and the tensions in the string on each side of the pulley. Determine the magnitude of the torque acting on the pulley in problem above. Lab Activity:Open the following link for a simulation of the rotation motion lab: https://ophysics.com/r5.htmlSelect the “Falling Mass” simulation and set the initial parameters for the simulation of Run the simulation and observe the values for the net Torque on the pulley, the acceleration of the mass, and the angular acceleration of the pulley. Compare your observations to your calculations in parts 1a-d. If the magnitudes of the values do not match closely, review your calculations in the previous section. Is the arithmetic sign of the torque and angular velocity in agreement with the convention that we are using in the textbook? Repeat the simulation for a solid cylinder and for a solid sphere using the same radius and mass of the rotating object and the same falling mass. Which object (cylindrical shell, solid cylinder, or solid sphere) has the largest angular acceleration. Determine for each rotating object (cylindrical shell, solid cylinder and solid sphere) the rotational inertia about the axis through the center of mass. Use this in explaining the results for the angular acceleration of the pulley in each case. For this simulation, let’s see if the timing is correct. Set the rotating object back to the cylindrical shell. Using a stop watch on your phone or watch, measure the time it takes the cylindrical shell to rotate one-half revolution (radians). Given the angular acceleration of the pulley and the kinematic expressions for constant angular acceleration, determine the time it should have taken for the angular displacement to be Is the simulation in real time? Or in slow motion? Select the “Two Masses” simulation, set the initial parameters to be left suspended mass is the right suspended mass is , and the pulley is a thin-walled cylindrical shell of mass and radius . Run the simulation and compare the resulting magnitude of the net torque, angular acceleration of the pulley and linear acceleration of the masses with your calculations from 1.e-g. Reflection on the resultFor the case in Activity 2a, the cylindrical shell rotates through one-half revolution during the run. Given the radius of the pulley, how far as the suspended mass dropped? Determine the potential energy change for the system. Determine the time interval it takes for the pulley to rotate that half-revolution (from the kinematics of the rotation). Determine the final total kinetic energy of the system of the pulley and the suspended mass at the end of the run. Determine the change in the kinetic energy of the system. Is the total mechanical energy of the system conserved in this simulation? Justify your answer. Conclusion: Conclusion:
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