General Physics IGeneral Physics I

Lecture 3: Newton's LawsLecture 3: Newton's Laws

What Causes Changes of Motion?What Causes Changes of Motion?

● We define the interaction of a body with its environment in terms of a force F.

● We begin by treating physical objects as particles with mass m.

● Laws of motion (how an object moves under a force) and the force laws (how to calculate the force).

What Causes Motion?What Causes Motion?

● No external forces is needed to keep a body move with constant velocity.

● There is no distinction between a body on which no external force acts and a body on which the sum or resultant of all the external forces (the net force) is zero.

Newton's First Law of MotionNewton's First Law of Motion

● In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity (that is, with a constant speed in a straight line).

● In simpler terms, when no force acts on an object, the acceleration of the object is zero.

● Any isolated object (one that does not interact with its environment) is either at rest or moving with constant velocity.

Newton's Second Law of MotionNewton's Second Law of Motion

● The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Newton's Third Law of MotionNewton's Third Law of Motion

● If two objects interact, the force F12 exerted by object 1 on object 2 is equal in magnitude to and opposite in direction to the force F21 exerted by object 2 on object 1:

● The action force is equal in magnitude to the reaction force and opposite in direction.

Significance of Newton's 1st LawSignificance of Newton's 1st Law

● Identifies a set of special reference frames in which we can apply the laws of classical mechanics.

● The tendency of an object to resist any attempt to change its velocity is called the inertia of the object.

● Newton's first law is often called the law of inertia. The reference frames to which it applies are called inertial frames.

Free Fall vs Projectile MotionFree Fall vs Projectile Motion

Q: If you want to calculate the horizontal speed of the yellow ball, is it enough that I tell you the time between shots was 1/30 second?

Relative MotionRelative Motion

● Now we choose the inertial frame that travels at the initial horizontal velocity of the yellow ball.

– The yellow ball is in free fall.– The red ball is in projectile motion, with the

horizontal velocity to the left. ● This is an example that all inertial observers agree

on measurements of acceleration, although they disagree on measurements of positions and velocities.

Relative Motion: Another ExampleRelative Motion: Another Example

Galilean Transformation Galilean Transformation

The two inertial observers agree on measurements of acceleration.

Where to Aim?Where to Aim?

r⃗P=v⃗0P t+12g⃗ t2 r⃗T=r⃗ 0T+

12g⃗ t2

Motion in Accelerated FrameMotion in Accelerated Frame

In a Rotating SystemIn a Rotating System

Newton's Second Law:

T=mv2

rT−m

v2

r=0

Mass Mass ≠≠ Weight Weight

● You can be weightless but not massless.

● Weightlessness in an orbiting spacecraft is physically identical to free-fall, with the difference that gravitational acceleration causes a net change in the direction, rather than the magnitude, of the spacecraft's velocity.

Application IApplication I

frictionless Acceleration? Tension in the cord?

I: Two Connected ObjectsI: Two Connected Objects

II: The Conical PendulumII: The Conical Pendulum

The ball lives in a three-dimensional world. The constancy of theta reduces the problem to a two-dimensional one.

However, there is only one degree of freedom for the circular motion of the ball, namely, the angle in the horizontal plane. There must be an additional constraint.

E.g., the ball cannot move at any speed given r fixed.

II: The Conical PendulumII: The Conical Pendulum

Application to Dynamical ProblemsApplication to Dynamical Problems

● Select a suitable inertial reference frame. Choose a coordinate system.

● Analyze all the forces applied on each object in the problem.

● For each body, find the vector sum of all the forces and apply Newton's second law to find the acceleration of that body.

● Solve for the velocity and the position for each body.

Pairs of ForcesPairs of Forces

● In all cases, the action and reaction forces act on different objects.

III. Drag (Fluid Resistance)III. Drag (Fluid Resistance)

● The magnitude of the resistive force can depend on speed in a complex way.

A) The resistive force is proportional to the speed of the moving object; this assumption is valid for objects falling slowly through a liquid and for very small objects, such as dust particles, moving through air.

B) The resistive force that is proportional to the square of the speed of the moving object; large objects, such as a skydiver moving through air in free fall, experience such a force.

R=cv2

R=bv

III.A: Terminal SpeedIII.A: Terminal Speed

Differential equation:

III.A: Terminal SpeedIII.A: Terminal Speed

● Static solution:

● Characteristic time constant t = m / b = vt / g:

● Acceleration:

a =dvdt

= (vt / τ)e−t / τ

= ge−t / τ

Ex: Sphere Falling in OilEx: Sphere Falling in Oil

τ =vt

g=

5.00 cm/s9.80 m /s2

= 5.10×10−3 s

vt (1−e−t / τ )=0.9vt

t = 2.30 τ = 11.7 ms

III.B: Drag at High SpeedIII.B: Drag at High Speed

r: density of fluidA: cross-sectional area of

the falling objectD: drag coefficient

ComparisonComparison

R =12DρAv2R = bvDrag:

Terminal velocity:

Low speed High speed

Check out the cool video by NASA on drag at http://en.wikipedia.org/wiki/Drag_%28physics%29

Scaling: A Powerful Concept Scaling: A Powerful Concept

● How does the terminal speed depend on the dimensions of the object?− Suppose the object is a sphere of radius r.− The cross-sectional area of the object scales as

A µ r2.− The mass of the object m scales as r3.− Terminal speed vt µ r1/2.

Are you smarter enough today?Are you smarter enough today?