This property is central to the utility of the momentum Model, as we shall see in the following modules. ![]() Only external forces can have a net influence on the motion of the system. The internal forces cancel out and have no net effect on the overall motion of a multi-body system (though as we shall see, it is important to remember that the motion of the pieces of a multi-body system are affected by internal forces). It is very important to notice the qualifier "external" on the sum of forces in this law. You can show that this form of Newton's Second Law reduces to the regular F = ma for the case of a single object with constant mass if you substitute in the definition of momentum in terms of velocity. In order to understand how interactions change momentum, we need a version of Newton's Second Law that involves momentum. In other words, the momentum Model will allow us to consider a single system to be composed of a collection of several point particles. Other than linear momentum, there is one another type of momentum, and that is angular momentum. The simplicity of this expression allows us to continue to think intuitively in terms of the components of the multi-body system, since it is easy to recover the momentum of the whole from the momentum of the parts. Linear momentum has a definite role in determining the 2nd law of motion. It turns out that the appropriate momentum to use is simply the sum of the momenta of the parts. Suppose instead we asked for the momentum of the system composed of the truck plus its cargo. Working with the motion of the center of mass is often counter-intuitive, and therefore we will not focus on it. This point moves in a way that is different than the cargo's movement and also different than the truck's. In other words, we would have to imagine the system to be a single point particle which is located at a special point somewhere between the position of the cargo and that of the truck (remember that the velocity Model can only be applied to a single point particle). What velocity should we assign to the system? It turns out that the appropriate velocity to use in the equations of dynamics is the velocity of the center of mass. Imagine you want to consider the truck plus the cargo as a single system. Unfortunately, the cargo is not well secured, and as the truck begins to accelerate the cargo starts to slide back along the truck bed.Īt a certain instant, the x-velocity of the truck is v t,x and the x-velocity of the cargo is v c,x. To see the advantage, consider this example:Ī truck of mass m t is loaded with a heavy cargo of mass m c. ![]() ![]() For a multi-body system, however, there is a significant advantage in using momentum instead of velocity to describe the motion. Momentum is Additive in a Multi-Body Systemįor a single object, momentum and velocity are essentially equivalent since momentum is directly proportional to velocity. The units of momentum, as can be seen from its definition, are kg m/s. The direction of the momentum of an object is exactly the same as the direction of the object's velocity. It is important to note that momentum is a vector. Momentum (which is given the symbol p) is defined by:įor a single object of mass m traveling with velocity v. p momentum mv v 1 speed of ball just before collision v 2 speed of ball just after collision u speed of the board just after collision The collisions between the inelastic and elastic balls and the board are partially inelastic collisions. ![]() Properties of Momentum Definition in terms of Velocity Explain the advantage of momentum over velocity as the description of motion in a multi-body system.Δ L \Delta L Δ L delta, L is change of angular momentum, τ \tau τ tau is net torque, and Δ t \Delta t Δ t delta, t is time interval.Ĭhange in angular momentum is proportional to average net torque and the time interval the torque is applied.In this module, we define momentum and impulse.Īfter completing this module you should be able to: Δ L = τ Δ t \Delta L=\tau \Delta t Δ L = τ Δ t delta, L, equals, tau, delta, t L L L L is angular momentum, m m m m is mass, v v v v is linear velocity, and r ⊥ r_\perp r ⊥ r, start subscript, \perp, end subscript is the perpendicular radius from a chosen axis to the mass's line of motion.Īngular momentum of an object with linear momentum is proportional to mass, linear velocity, and perpendicular radius from an axis to the line of the object's motion. L = m v r ⊥ L=mvr_\perp L = m v r ⊥ L, equals, m, v, r, start subscript, \perp, end subscript L L L L is angular momentum, I I I I is rotational inertia, and ω \omega ω omega is angular velocity.Īngular momentum of a spinning object without linear momentum is proportional to rotational inertia and angular velocity. L = I ω L=I \omega L = I ω L, equals, I, omega
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