Contact forces

The atoms (原子) in a solid material can be thought of as balls connected by springs (弹簧). The ‘springs’ are actually electromagnetic forces between the electrons and protons in the atoms.

Normal force 正压力 When something compresses the springs, they push back with enough force to balance the compressing force.

Normal force FN ‘Normal’ is another word for ‘perpendicular’ (垂直). The direction of the normal force is normal to the surface.

Friction 摩擦力 Friction is caused by interactions between the atoms in two surfaces trying to slide past each other.

Friction has two types Static friction: when the two surfaces do not slide Kinetic friction: when the two surfaces do slide The direction of the friction force is opposite that of the force trying to make the object slide.

μs and μk are called the ‘coefficients of friction’. Friction has two types μs and μk are called the ‘coefficients of friction’.

Example I push a book against the wall with a force of 30 N. The mass of the book is 3 kg. The coefficient of static friction for the book and the wall is 0.6. Will the book slide?

Example A car of mass m = 1600 kg moves at a constant speed around a flat, circular track of radius R = 190 m. The value of μs between the track and the tires is 0.20. What is the largest speed v the car can have without sliding off the track?

Tension 张力 The force at each end of a rope has the same magnitude, and is directed along the rope.

Calculus and Newton’s 2nd Law Sometimes, we want to find a formula for the velocity (or position) of an object as a function of time. 公式

(1) Solve Newton’s 2nd Law for the acceleration.

(2) Find the velocity by integrating the expression for the acceleration. 求它的积分

(2) Find the velocity by integrating the expression for the acceleration.

(2) Find the velocity by integrating the expression for the acceleration.

(3) If necessary, integrate again to find the position.

Example What is the least speed for a projectile launched vertically upward from the Earth’s surface, such that it will continue into space and never return to Earth? Assume that gravity is the only force felt after launch.

GIVEN: Mass of the object m Mass of the Earth mE Height of the object from the center of the Earth y(t) Gravitational force Fg = GmEm/y2 (down) FIND: Initial speed vi such that the object never returns to the Earth.

PROCEDURE: Use Newton’s 2nd Law to find the acceleration of the object. Integrate the acceleration to get an expression for the speed as a function of time. Analyze this expression to find the minimum launch speed such that the velocity never changes direction.

Example When a small sphere, starting from rest, falls through a liquid, it experiences a drag force FD in addition to the force of gravity. FD is directed upwards and has a magnitude FD = bv, where b is a constant. Find the speed of the sphere as a function of time, v(t).

GIVEN: Mass of the object m Weight force Fg = mg (down) Drag force Fd = bv (up) FIND: Velocity v(t)

PROCEDURE: Use Newton’s 2nd Law to find the acceleration of the object. Integrate the acceleration to get an expression for the velocity as a function of time.

It is nice to solve Newton’s 2nd Law with calculus… but usually it doesn’t work! Example: The motion of three planets, interacting via gravity, has no analytical solution.

In such cases, we must solve Newton’s 2nd Law numerically, using computers.