Systems of Particles & Linear Momentum - Unit 4

Systems of Particles & Linear Momentum - Unit 4 Linear motion of an object is usually thought of as displacement, velocity and acceleration of its center of mass. When you draw a free-body diagram, for example, you often have all of your forces represented as acting on a single point, which is the center of mass. Where is the Center of Mass? The center of mass in a symmetrical, regular solid of uniform mass density is at its geometric center. For a non-uniform solid, we have to think a little deeper. If we have a collection of point masses, then the center of mass of that collection of masses is: 𝑥𝑐𝑚 = ∑ ∑ 𝑚𝑖 𝑥𝑖 𝑚𝑖 Essentially, it is the sum of the products of the masses and positions of each particle, divided by the sum of the masses of the particles. Center of Mass (Calculus Definition) The calculus definition of the center of mass is as follows: 𝑥𝑐𝑚 = ∫ 𝑥𝑑𝑚 ∫ 𝑑𝑚 Movement of the Center of Mass If there is no net force acting on an object or system, then the center of mass won’t accelerate. This means that if, for example, an object explodes while moving (an internal interaction and internal force) and splits into 2 objects, then the center of mass of the 2 fragments will move at the same velocity it was going at. And there is a special case for this concept: if there is a force providing the same acceleration to every single fragment, like gravity, then if you, for example, launch an object that exploded in mid-air, the center of mass of the fragments would continue in the same path that the object originally would have taken. Linear Momentum Any object moving with velocity has momentum defined by: 𝑝 = 𝑚𝑣 The total momentum of a system of objects is the vector sum of the momenta of each object. The rate of change of the momentum of a system is equal to the net external force. 𝑑𝑝 𝐹= 𝑑𝑡 We can derive the Second Law of Motion from this with a little work. 𝐹= 𝑑𝑝 𝑑 (𝑚𝑣) 𝑑𝑣 = =𝑚 = 𝑚𝑎 𝑑𝑡 𝑑𝑡 𝑑𝑡 Impulse Impulse is the average force acting on an object over a time interval. 𝐽 = 𝐹𝑎𝑣𝑔 𝛥𝑡 Impulse is also equivalent to the change in momentum of the object receiving impulse. 𝐽=∫ 𝐹𝑑𝑡 = 𝛥𝑝 As you can see from the relationship, impulse (and therefore change in momentum) can be calculated as the area underneath a force-time graph. Collisions When a system of objects isn’t acted upon by any external forces, linear momentum is conserved. You can use this conservation of momentum in collision situations, sometimes alongside conservation of energy, to find out more about how things are moving. This is a result of the fact that forces internal to a system won’t change the momentum of the center of mass of the system (the sum of the momenta of each component). You can use this to your advantage. For example: if you have 2 pool balls going at a certain velocity that collide, and you know the direction + velocity that one of the balls after the collision, then you can find the second ball’s velocity and direction by equating the sum of the momentums before and after. Elastic Collisions Elastic collisions are a special type of collision, where both momentum and kinetic energy are conserved. Regular collisions are usually not elastic because they lose some energy in the collision: the sound that things make when they collide, for example, is energy that is emitted through sound. Elastic collisions occur usually when there is something that stores energy and then releases it during the collision, such that the energy isn’t lost in the collision. For example, springs on 2 carts that compress when they touch, eventually fully expanding to release all kinetic energy back into their motion, rather than losing some energy. Completely Inelastic Collisions There is one last type of special collision: this is the completely inelastic collision, where the maximum amount of kinetic energy is lost. This is usually when the 2 objects in the collisions stick together and move with the same velocity, resulting in the most energy lost.