Newton’s Laws of Motion - Unit 2

Newton’s Laws of Motion - Unit 2 Newton’s First Law An object at rest or in uniform motion will remain at rest or in uniform motion unless acted on by a net external force. Uniform motion means constant velocity. “At rest” just means that an object is in uniform motion, but with v = 0. As long as an object is moving in uniform motion, it must have 0 net external force acting on it (Fnet = 0). This means all directions. For example: - If you have a hockey puck sliding on smooth and slippery ice, the gravity and normal force with the ice will cancel to create zero net force, and the puck will keep on sliding with constant velocity. - If you have a spaceship in deep space that isn’t close to any stars or planets, there will be no (or negligible) forces acting on it, and it will just keep on going at a constant velocity. Do note, however: the mass of the object must be constant. This year, I had a free response question with a moving cart that had changing mass: it’s possible that you could get changing mass in an exam problem, too. The Second Law of Motion The sum of the forces acting on an object is proportional to its mass and its acceleration. 𝐹=∑ 𝐹 = 𝑚𝑎Note: this is actually a special case that assumes that mass is constant. If mass isn’t constant, then you need to do something a little more clever: that’s outside of this unit, however. The Third Law of Motion For every action, there is an equal and opposite reaction. For every action force on an object B due to another object A, there is a reaction force which is equal in magnitude but opposite in direction on object A due to object B. 𝐹𝐴 𝑜𝑛 𝐵 = −𝐹𝐵 𝑜𝑛 𝐴 Remember: the action and reaction forces are acting on different objects. Forces Forces are interactions between 2 objects: they are pushes or pulls. There are 2 categories of forces: contact forces, which act between objects that are in contact, and non-contact forces that act between objects without them touching. Equilibrium If the net force on an object is zero, then the object is in a state of equilibrium. There are 2 types of equilibrium: - Dynamic equilibrium is when the object is moving relative to us (the observers) - Static equilibrium is when the object is not moving relative to us. Normal Force Normal force is the reaction force that prevents objects from going into each other. When 2 objects are in contact and pressing into each other, there will be normal force preventing, say, you from falling through the floor, or your cup from falling through the table. Friction Force Friction is a force that opposes the sliding of two surfaces against one another. It always acts in a direction that opposes motion or attempted motion. Frictional force is dependent on: - Normal force N, or the force of the two surfaces pressed against each other - Coefficient of friction, either static or kinetic, which is dependent on smoothness of the surfaces and any lubricants used. Static Friction Static friction between two surfaces is when there is no relative motion between them. It increases to match any applied force, just enough to prevent the two surfaces from moving relative to each other, and is at maximum when the object is just about to move. 𝑓𝑠 ≤ 𝜇𝑠 𝑁 𝜇𝑠 is the coefficient of static friction, and N is the magnitude of the normal force. Kinetic Friction Kinetic friction occurs between two surfaces when they are moving relative to each other. Kinetic friction is constant as long as the normal force stays constant. 𝑓𝑘 = 𝜇𝑘 𝑁 𝜇𝑘 is the kinetic friction coefficient, and again, N is the magnitude of normal force. Applying the Laws: Pulleys and Atwood Machines Consider this pulley system. Assuming that the tabletop is frictionless and that the weight of the pulley and the string is negligible, how do we determine the acceleration of the blocks? We have 2 blocks, so we need to draw 2 free-body diagrams, and use the Second Law to find acceleration. We get 2 different equations from these free-body diagrams + the Second Law, and we can add the equations to remove the tension force. We can then solve for acceleration. 𝑀𝑔 = 𝑀𝑎 + 𝑚𝑎 𝑀𝑔 = (𝑀 + 𝑚)𝑎 𝑎= 𝑀𝑔 𝑀+𝑚 Working out the relationships between the motion of different objects using Newton’s Second and Third Laws is a key idea of this unit, and will come up many times throughout the course. Think about how different objects interact with each other: what forces they will apply on each other, if their accelerations or velocities are linked (like with the pulleys), etc. Uniform Circular Motion If an object is travelling in a circle around a center point, it needs some specific conditions. The object MUST be travelling at constant speed (NOT velocity) around the center, and so much acceleration acting on it that is pointing to the center of the circular motion. This acceleration is called the centripetal acceleration. It is given by: 𝑣2 𝑎𝑐 = 𝑟 For example: If there is an object moving in a circle at a constant speed of 10 m/s and the motion has a radius of 2 meters, what is the centripetal acceleration? 𝑣 2 102 𝑎𝑐 = = = 50 𝑚/𝑠 2 𝑟 2 NOTE: Centripetal force isn’t an actual force. Centripetal force is the result of other forces. Some other force like tension or gravity must provide a force towards the center of the circular motion.