Newton’s Laws 1st Law: A body acted on by no net force moves with constant velocity (which may be zero) 2st Law: The acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the net force acting on the object. 3rd Law: For every action there is an equal, but opposite reaction

The First Law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. It may be seen as a statement about inertia, that objects will remain in their state of motion unless a force acts to change the motion.

Aristotle: a natural state of an object is at rest; a force is necessary to keep an object in motion. It follows from common sense. 384-322 B.C. Galileo: was able to identify a hidden force of friction behind common-sense experiments, abstracted from it a fundamental law of inertia and the principle of relativity 1564-1642

First Law is identical to Galilean Principle of relativity Galileo Galilei Galilean principle of relativity (Galileo’s ship!) Laws of physics look the same for all observers who move with a constant velocity with respect to each other, i.e. in all inertial frames of reference. 1564-1642 Indeed, if no force in needed to keep the body in motion with constant velocity, all such states of motion are equivalent. For different inertial observers the object will appear moving with different but constant velocity (which may be zero).

The First Law contains implications about the fundamental symmetry of the universe in that a state of motion in a straight line must be just as "natural" as being at rest. If an object is at rest in one frame of reference, it will appear to be moving in a straight line to an observer in a reference frame which is moving by the object. There is no way to say which reference frame is "special", so all constant velocity reference frames must be equivalent. The first law is valid only with respect to an inertial observer, i.e. in inertial frames of reference. It is violated in accelerated reference frames.

2nd Law From experiments we know: A force is needed to change the state of motion Force is a vector; obeys superposition principle: the net force is a vector sum of all forces acting on an object The direction of acceleration vector is the same as the direction of the force vector The magnitude of the force and acceleration are related by a constant which intuitively is a “quantity of matter”. This is the inertial mass.

Newton’s 2nd Law of Motion The acceleration a of a body is inversely proportional to its mass m, directly proportional to the net force F, and in the same direction as the net force. a = F/m  F = m a 1 N = 1 kg m/s2

Newton’s 3rd Law of Motion To every action, there is an equal and opposite reaction. The same force that is accelerating the boy forward, is accelerating the skateboard backward.

Clockwork universe Newtonian mechanics was so beautiful and clear that people believed that it explained everything. They thought that everything in the Universe worked like a perfect clock obeying Newton’s laws. To predict what happens in future, you just need to specify the initial positions and velocities for all objects.

Types of forces Gravity and weight Normal force Friction Tension and spring force Contact versus long-range forces All are manifestation of four Fundamental “forces” Gravity Electromagnetic Strong Weak

M Gravity is a strange force. It has a unique property: All bodies in the same point in space experience the same acceleration! Galileo, about 1600 m R M

Weight, “apparent weight”, the force of gravity, and the normal force Riding an elevator

Units of Force One pound is 0.4536 kg British system: units of mass: units of force: One pound is 0.4536 kg One pound is the weight of 0.4536 kg on the Earth

Box on an inclined plane A box with mass m is placed on top of a frictionless incline with angle q and height H and is allowed to slide down. What is the normal force? What is the acceleration of the box? What is the velocity of the box when it reaches the bottom? q

Kinetic: The friction force that slows things down Two types of friction: Kinetic: The friction force that slows things down Static: The force that makes it hard to even get things moving

Refrigerator If you push a refrigerator when there is no friction what happens? In the real world what happens? Especially when it’s fully loaded and on a sticky kitchen floor? When does static friction kick in? When does kinetic friction kick in?

Friction There is some maximum value the friction force can achieve, and once we apply a force greater than this maximum there is a net force on the object, so it accelerates. The maximum of the force of friction varied linearly with the amount that the block pushes on the table.  - coefficient of friction, is the vertical force exerted by the block on the table The friction force only exists when there is another force trying to move an object

FFriction = mKineticFNormal Kinetic Friction For kinetic friction, it turns out that the larger the Normal Force the larger the friction. We can write FFriction = mKineticFNormal Here m is a constant Warning: THIS IS NOT A VECTOR EQUATION!

FFriction  mStaticFNormal Static Friction This is more complicated For static friction, the friction force can vary FFriction  mStaticFNormal Example of the refrigerator: If I don’t push, what is the static friction force? What if I push a little?

Is it better to push or pull a sled? You can pull or push a sled with the same force magnitude, FP, and angle Q, as shown in the figures. Assuming the sled doesn’t leave the ground and has a constant coefficient of friction, m, which is better? FP FP

A Recipe for Solving Problems Sketch Isolate the body, draw a free-body diagram (only external forces but not forces that one part of the object exert on another part) 2. Write down 2nd Newton’s law Choose a coordinate system Write 2nd Newton’s law in component form: 3. Solve for acceleration

Pulling Against Friction A box of mass m is on a surface with coefficient of kinetic and static friction m. You pull with constant force FP at angle Q. The box does not leave the surface. Find the minimum force you need to apply in order to move the block What is the magnitude of the acceleration? What angle maximizes the acceleration? Q

Box on an inclined plane with friction A box with mass m is placed on an incline with angle q and is allowed to slide down. What is the acceleration of the box? q

Tension and pulleys

Force of tension Massless, unstretchable string; massless, frictionless pulley

The advantage of a pulley What minimum force F is needed to lift the piano of mass M?

Conical pendulum

A ball of mass m is swung around a circle at the end of a string of length L. The string will break if the tension in it exceeds a critical value, Tc. What is the largest constant angular velocity the ball can have without breaking the string? What is the largest period the ball can have without the string becoming slack?

A mass m1 is going around in a circle on a string on a frictionless table and the string goes through a hole where it is attached to a hanging mass m2. If the mass m1 is going around with constant , what must the distance from the mass m1 to the hole be if the mass m2 is to remain at rest? m1 m2

Playing with weight: A car on an arched bridge Your weight in a rotating space station or on the rotating Earth

A race track designer wants to have the cars able to maintain a speed vmax without skidding on a circular track. If the track is flat with a coefficient of friction  what does the radius have to be? A race track designer wants to have the cars able to maintain a speed vmax without skidding. At what angle must the track of radius R be banked assuming no friction? Assuming a coefficient of friction ?

A satellite of mass m is attracted to the Earth of mass M with a force of gravity proportional to the inverse square of the distance to the Earth center, r:  is a gravitational constant Find the velocity of a satellite on circular orbit of radius r. Find the radius of the orbit for a geostationary satellite