Lecture 7: More on Newton s Laws Other Important Aspects of the Second Law: Note that = ma is a vector equation, i.e., it is equivalent to saying: = ma x y z = ma = ma An object accelerates in the same direction as the net force applied to it Units of force: Dimensional analysis using the Second Law shows that force has dimensions of (mass)x(length)/(time) 2 In our standard units, 1 Newton = 1 kg m/s 2 x y z
Utility of the Second Law rom our study of kinematics, we know that if we are given an object s acceleration, initial velocity and initial position, we can determine its velocity and position at any later time So Newton s Second Law is a prescription for predicting the motion of an object if the forces acting on it are known
Newton s Third Law Objects feel forces due to their interaction with the environment e.g., if I push a book across the desk, the book feels forces from my hand, and frictional forces from the desk s surface The environment, in turn, also feels forces due to the object My hand feels that something is pushing back, or resisting its motion; the desk feels itself being pulled in the direction the book is moving Newton s Third Law tells us that any such pair of forces are equal in magnitude and opposite in direction
Note that Newton s Third Law concerns a pair of forces acting on different objects Not the same as the case where two forces acting on the same object happen to cancel Sometimes seems counter-intuitive: A bug hitting the windshield of a speeding car exerts a force on the car of the same magnitude that the car exerts on the bug But the car is able to withstand much larger forces than the bug, which accounts for the difference in effect on the two objects
Solving Problems Using Newton s Laws ollow these steps and you ll never go wrong: 1. Make sure you re in an inertial frame (or at least a good approximation to one, such as the surface of the Earth). Choose a convenient coordinate system e.g., if the motion is going to be in a particular direction, make that direction one of your axes 2. Draw a free-body diagram for each object whose motion you wish to study. You may treat the object as a particle, and show all the forces acting on it. Label these forces carefully. 3. ind the components (in your coordinate system) of all the force vectors acting on the object, and sum them to find the net force.
Example The captain of a 5000-kg spaceship wants to accelerate at 100m/s 2 in the direction shown. With how much thrust should each engine be fired? a θ = 37 o Engine 1 Spaceship Engine 2
Start by drawing a free-body diagram of the spaceship: y 1 2 x With this choice of axes, we can write the net force on the spaceship as: = i + j 1 2 We want to have the following properties: = m a = 5000kg 100m/s = 4 5 10 N 2 Direction 37 o from x axis
In terms of the components of : 2 2 4 1 2 o tan 37 0.75 Now we can solve for 1 and 2 : 1 2 + = 5 10 N = = 1 2 ( ) 2 2 1 = 0.75 2 2 4 2 + 2 = 0.75 5 10 N 1.57 = = 5.0 10 4 3.9 10 N = 2.9 1 4 0 N 4 N
Weight and Mass We intuitively feel that mass is correlated with weight Heavier objects are also more massive that is, they re harder to push around But weight and mass are different quantities Mass, as we ve seen, is a measure of an object s resistance to acceleration Weight, on the other hand, is the force exerted by gravity on an object In a spaceship very far from any other object, you would feel almost no force from gravity Your weight would then be near 0, but your mass would be the same as it is on Earth
Weight (near the Earth s surface) We observe that all objects fall at the same rate if gravity is the only force acting on them i.e., gravity imparts the same acceleration to all objects We need to reconcile this with Newton s 2 nd Law: = ma a = m It must be that the force due to gravity the weight -- has the form: W = mg = mgj (If we take the y axis to be vertical) The constant g is 9.8 m/s 2