- Physics Laboratory Exercise Conservation Of Linear Momentum And The Ballistic Pendulum Introduction The Principle Of Con 1 (660.76 KiB) Viewed 83 times
a table and allowing it to fall freely toward the earth, the initial speed can be determined from the projectile’s range and its vertical displacement during the fall. Method I: Determing v Using Collision Physics The momentum of a body is defined as the product its mass and its instantaneous velocity: P=mỹ. In its most general form, Newton's second law of motion states that the net force acting on a body is proportional to the time rate of change of momentum: Fnet = dø/dt This form of Newton's second law is more general than the commonly recognized version CÈDet mà) because it does not assume that the mass is constant. In fact, it is easy to show that Newton's second law in the special case of constant mass (niet = mā) follows directly from the general form of the second law et = dp/dt) simply by plugging in the definition p=mỹ and taking the derivative. Now, if the sum of the external forces acting on a system is zero (if net = 0) it follows directly that the linear momentum of the system is constant: 0 = dp/dt const=P This is the principle of conservation of linear momentum. Applied to a system of bodies (such as two or more colliding objects), the principle implies that if no net external force acts on the system then the momentum of the system's center-of-mass does not change. During a collision between two bodies, each exerts a force on the other. But by Newton's 3rd law these forces are equal and opposite, so if the two colliding bodies are treated as a single system these internal action/reaction pairs cancel out. Thus, while the individual momenta of the colliding bodies do change as a result of the collision, the total momentum of the system (which is the vector sum of the individual momenta) is not changed by the impact. Hence the total momentum of the system after the collision is equal to the total momentum of the system before the collision. During a collision the interacting bodies may be deformed, and if so energy is consumed to change their shape. If the bodies are highly elastic, however, they may recover completely from the distortion and will return all of the energy used in distorting them. In this special case, the case of elastic collisions, the total kinetic energy of the system remains constant. If the bodies are not sufficiently elastic, or if they are locked together following the collision, the system will remain permanently distorted and the energy used in producing the distortion is not recovered. Such 2
Histófs are carrea either inelastic or perfectly inelastic, depending on whether the objects stick werner to the perfectly inelastic case the objects stick together after the collision, and significant energy is used in forming the bond between them; with inelastic collisions, energy is lost in deforming the objects (and perhaps also in producing heat and sound during the impact) baxt the objects do not stick together. Perfectly inelastic impacts occur in the ballistic pendulum, which is the device we will employ in this lab. If a projectile ball is fired into a pendulum bob and remains embedded in it, the momentum of the bob and ball just after the collision is equal to the momentum of the bob and ball just before the collision. This follows from the law of conservation of momentum. The velocity of the pendulum before the collision is zero, while after the collision, the pendulum and the ball move with the same velocity. Hence, conservation of momentum gives the equation (1) My = (m+MV where m is the mass of the ball, vis the velocity of the ball just before the collision, Mis the mass of the pendulum bob, and V is the common velocity of the bob and ball just after the collision. Unfortunately, however, in this single equation we have two unknowns: V, which we need to find, and the common velocity V of the ball and bob just after the collision. Fortunately, we can use the motion of the system after the collision to find V (see figure below). Pivot 10. I M Ah m After the collision the pendulum and ball swing up, rising through a vertical distance Ah. From a measurement of this distance Ah it is possible to calculate the velocity V. The kinetic energy of the system just after the collision must be equal to the increase in potential energy of the system as the pendulum reaches its highest point. This follows from the law of conservation of energy, assuming that the loss of energy due to friction is negligible. Thus, conservation of energy equation applied to the swinging motion after the collision gives
where M is the mass of the pendulum bob, m is the mass of the ball, V is the common velocity of the pendulum bob and ball just after the collision, g is the known value of the acceleration of gravity, and sh is the vertical distance through which the center of gravity of the system rises. The left side of the equation represents the kinetic energy of the system just after impact; the right side represents the change in the potential energy of the system during the swing. Solving this equation for V, one obtains: (3) T = v2gah By substituting this value of V and the values of Mand m into the momentum equation (equation (1)), it is possible to calculate the velocity of the ball just before the collision. The apparatus used in this experiment is a combination of a ballistic pendulum and a spring gun. The pendulum is a heavy cylindrical bob (hollowed out to receive the projectile) which is suspended by a strong light rod. The rod is pivoted at its upper end to allow the system to swing upward after the collision. The projectile is a brass ball which is fired into the pendulum bob and is held there by a rubber O-ring. A ratcheting system catches the bob near the peak of its swing by engaging a tooth in a curved track, and a numerical index attached to the frame is used to measure the maximum height reached by the loaded pendulum (see figure below). Figure: Close-up of peridulum bob caught near peak of swing by ratcheting system.
Method II: Confirming v Using Projectile Motion The velocity of the ball can also be confirmed using the physics of projectile motion. In this method, the pendulum is first locked out of the way so that the ball can be fired horizontally from a table top. Measurements of the projectile's range and vertical displacement then lead directly to v. Projectile motion is the two-dimensional case of a freely falling body in which the initial velocity may be in any direction. The path of the projectile is a parabola produced by a combination of the uniform horizontal velocity and the vertical velocity, which changes due gravity. This problem is solved by treating it as two independent motions: one with constant velocity in the horizontal direction, and the other of constant acceleration in the vertical direction. The problem is also simplified by neglecting the effects of air friction, which is a reasonable approximation when the projectile speed is relatively low and the distances traveled are small. Projectile motion is described by the by the kinematic equations of motion. If air friction is neglected the horizontal component of the velocity remains constant, so the displacement in the horizontal direction (the range R) is given by the horizontal kinematic equation (4) R = x - Xo = (vaxt = vt where x - xo is the displacement of the projectile in the horizontal direction, v=(v.)x is the initial horizontal velocity, and t is the time. If the ball is fired horizontally, the initial velocity in the y direction is zero. Hence the motion in this direction is the same as that of a freely falling body with zero initial velocity. Therefore the vertical displacement is given by (5) y-yo (Velyt – 1/2gt2 = - 12gt2 where y-y, is the vertical displacement, g is the known value of the free-fall acceleration due to gravity, and t is the time of flight. After using algebra to eliminate the flight time t between equations (4) and (5) and then measuring x -- X, and y-y, directly in the lab, v is easily determined. Materials and Apparatus 육 1. Ballistic pendulum 2. Triple beam balance 3. Meter sticks) 4. Plastic see-through ruler 5. Plumb bob 6. Carbon paper 7. Torpedo level
Procedure I: Ballistic Pendulum Method (1) Record your device's ID code on the label taped to the base of your pendulum: V2 (2) Use a level to make sure the apparatus sits horizontally. Release the pendulum from the track and allow it to hang freely. Make sure the bob and ball are aligned, so the ball will stick inside after impact. Also, lightly tighten the pivot screw before EVERY trigger pull! (3) Prepare the gun for firing by inserting the ball on the end of the spring gun and pushing it back, compressing the spring until the trigger is engaged. (4) Make sure the pendulum bob is completely at rest. The track where the pendulum will be caught contains numbered slots. Fire the ball into the cup and (if it sticks) record the slot number on the track where the prawl is caught at the highest point. (5) Repeat steps (2) through (4) until you have completed a total of five trials Data: Ballistic Pendulum Method Position Trial #1 Trial #2 Trial #3 Trial #4 Trial #5 Average Slot Number 26 28 27 126 28 27. Table 1 (6) With the pendulum hanging in its lowest position, measure and record (below) the vertical distance hy from the base of the apparatus to the index point (the sharp "tip" on the bob). (7) Set the pendulum in the slot number which corresponds most closely with the average reading. Measure and record the vertical distance hy from the base of the apparatus to the index point attached to the pendulum. Initial height of Pendulum (h) 0.5 cm -9.65cm Average final height of Pendulum (ha) 16.15 cm 42-hash 10x15cm-uisem Change in height during swing (Ah) _0.650mg 6
Data: Projectile Motion Initial Horizontal Position (X.): Hei 13 cm Height of bottom of ball above floor: 83.3 cm Vertical Displacement of ball (y - yo), assuming floor level is at y=0: Distance Trial #1 Trial #2 Trial #3 Trial #4 Trial #5 2811 รา Final Position (x) 76.4 74.25 72.7 Range (x - x) 61.75 60.05 60.05 57.9 56035 Table 2 Average range R 59.34 Time-of-flight t 0.4125 V. - 5.89 ms
(4) Starting with the general form of Newton's second law (Fnet = dp/dt), derive the commonly- recognized version of the second law Înet mà). What assumption must be made here? (5) Calculate the kinetic energy KEinitial of the ball just before impact using your results from the ballistic pendulum (collision method). (6) Calculate the kinetic energy of the pendulum bob and ball just after impact from the speed V of the combined system just after the collision and the object masses. (7) Calculate the energy lost in the collision, Eloss ·