Figure 9.2 A graph of a non-constant force with respect to time. Figure 9.4 A diagram of a cart as a system and its surroundings Question: Suppose you push a baseball bat lying on a nearly frictionless table at the center of mass with a force . Will the acceleration of the center of mass be greater than, equal to, or less than if you push the bat with the same force at one end? Definition: Conservation of Momentum I. Understand – get a conceptual grasp of the problem. II. Devise a Plan - set up a procedure to obtain the desired solution.Chapter 9 Momentum Law II: The change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed. If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force is impressed altogether and at once or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both. Isaac Newton Principia Introduction When we apply a force to an object, through pushing, pulling, hitting or otherwise, we are applying that force over a discrete interval of time,tΔ. During this time interval, the applied force may be constant, or it may vary in magnitude or direction. Forces may also be applied continuously without interruption, such as the gravitational interaction between the earth and the moon. In this chapter we will investigate the relationship between forces and the time during which they are applied, and in the process learn about the quantity of momentum, the principle of conservation of momentum, and its use in solving a new set of problems in mechanics: collisions. 9.1 Momentum (Quantity of Motion) Newton defined the quantity of motion or the momentum, pG, to be the product of the mass and the velocity m=pvGG. (9.1.1) Momentum is a vector quantity, with direction and magnitude. The direction of momentum is the same as the direction of the velocity. The magnitude of the momentum is the product of the mass and the instantaneous speed. Units: In the SI system of units, momentum has units of 1[kg m s ]−⋅⋅ . There is no special name for this combination of units. 8/25/2008 1Average Force, Momentum, and Impulse Suppose you are pushing a cart with a force that is non-uniform, but has an average value during the time interval aveFGtΔ. We can find the average acceleration according to Newton’s Second Law, ave avem=FaGG. (9.1.2) Recall that the average acceleration is equal to the change in velocity divided by the time interval , ΔvGtΔ avetΔ=ΔvaGG. (9.1.3) Therefore Newton’s Second Law can be recast as ave avemmtΔ==ΔvFaGGG. (9.1.4) The change in momentum is the product of the mass and the change in velocity, mΔ=ΔpvGG. (9.1.5) Newton’s Second Law can be restated as follows: the product of the average force acting on an object and the time interval over which the force acts will produce a change in momentum of the object, ave.tΔ=ΔFpGG (9.1.6) This change in momentum is called the impulse, ave.t=Δ=ΔIF pGGG (9.1.7) Force is a vector quantity; impulse is obtained by multiplying a vector by a scalar, and so impulse is also a vector quantity. The SI units for impulse are -1[N s] [kg m s ]⋅=⋅⋅, which are the same units as momentum. Non-Constant Force and Impulse Suppose you now let a cart roll down an inclined plane and bounce against a spring mounted at the bottom of the inclined plane (Figure 9.1). The spring is attached to a force 8/25/2008 2sensor. The force between the spring and the cart is a non-constant force, , applied between times and()tFG0tft. Figure 9.1: Cart sliding down inclined plane and colliding with a spring at the base and reversing motion In Figure 9.2, we show a sample graph of force vs. time for the cart-spring system as measured by the force sensor during the time the spring is compressed by colliding with the cart. Figure 9.2 A graph of a non-constant force with respect to time. If we divide up the time interval into parts, then the impulse is approximately the vector sum of the impulse for each interval, N 1.iNiiit==≅Δ∑IFGG (9.1.8) The total impulse is the limit of this sum as we take smaller and smaller intervals. This limit corresponds to the area under the force vs. time curve, IG 8/25/2008 3(9.1.9) 001lim ( ) .fittiNiititttt==Δ→===Δ≡∑∫IFFGGGdtdt Since force is a vector quantity, the integral in Equation (9.1.9) is actually three integrals, one for each component of the force. Using Equation (9.1.7) in Equation (9.1.9) we see that ()()00() .fttftttt t==Δ= − =∫pp p FGGG G (9.1.10) The Fundamental Theorem of Calculus, applied to vectors, then gives ()dtdt=pFGG (9.1.11) for both constant and non-constant forces. Equation (9.1.11) is also obtained by taking the limit in Equation 0tΔ→ (9.1.4). In using either expression, it must be assumed that the mass of the object in question does not change during the interval tΔ. 9.1.1 Example: Impulse for a Non-Constant Force Suppose you push an object for a total time 1.0 stΔ= in the x+-direction. For the first half of the interval, you push with a force that increases linearly with time according to ()1ˆ( ) , 0 0.5s with 2.0 10 N stbt t b1−=≤≤ =×FiG⋅. (9.1.12) Then for the second half of the interval, you push with a linearly decreasing force, (9.1.13) ()1ˆ( ) , 0.5s 1.0s with 2.0 10 Ntdbt t d=− ≤≤ = ×FiG The force vs. time graph is shown in Figure 9.3. What is the total impulse applied to the object? 8/25/2008 4Figure 9.3 Graph of force vs. time Answer: We can find the impulse by calculating the area under the force vs. time curve. Since the force vs. time graph consists of two triangles, the area under the curve is easy to calculate and is given by () ()()()211 2 111ˆ( /2) ( /2)2211ˆˆ(2.0 10 N s )(1.0s) 1.0 10 N s .22bt t bt tbt−⎡⎤=ΔΔ+ΔΔ⎢⎥⎣⎦=Δ= × ⋅ =× ⋅IiiiGˆi (9.1.14) 9.2 External and Internal Forces and the Change in Momentum of a System So far we have restricted ourselves to considering how the momentum of an object changes under the
View Full Document
Unlocking...