Physics MWCC Prof. E. Stevens Griffith, W. Thomas & Brosing, Juliet W. The Physics of Everyday Phenomena. New York: McGraw Hill, 2009.

### Centripetal Force

Force(s) that make a body follow a curved path.

It is directed orthogonally to the velocity of the body, toward the instantaneous center of curvature of the path.

### Centripetal Acceleration

Rate of change of tangential velocity

Text: rate of change in v of an object that is associated with the change in direction of the velocity. It is always perpendicular to the v vector and toward the center of the curve.

Example: Whirling a ball on a string

The string acts to pull the ball inward. Without it, the ball will fly off in the direction of travel at the point the string breaks. Gravity pulls it downward. The string changes the v vector continually.

### Find v & centripetal

v₂-v₁=∆v

Each of the three will have a different direction

If the time span is short enough, ∆v will point toward center of curve

a(c)=v²/r

a sub c indicates that the acceleration vector is toward center of curve

Speed directly related to ∆v length

Speed directly related to direction of v vector change

Radius of curve inversely related to change in velocity

### Centripetal Acceleration Cause

Ball on string has both horizontal and vertical components since string isn't entirely within plane of motion

Horizontal component pulls ball toward center of horizontal circle producing centripetal acceleration

Total tension on string determined by both H and V components

V component is equal to w of ball, therefore vertical Fnet=0; thus the ball stays in the horizontal plane of the circle and does not accelerate vertically

centripetal acceleration proportional to square of speed of ball

### Centripetal equations

a_c=v²/r; v²=a_c*r; r=v²/a_c

Proportional to square of speed, inversely proportional to radius of curve

F_net=T_h=ma_c

net force=horizontal tension=mass*centripetal acceleration

F_net=mv²/r

net force = mass * velocity squared / radius

### Centripetal Force

Any force or combination of forces that act on an object to produce centripetal acceleration

Can be caused by: pull from string, push from contacting another object, friction, gravity, etc.

### Coefficient of friction

μ Greek mu

static or kinetic is denoted with subscript s or k

static: no motion in direction of force

kinetic: motion in direction of force

static usually > kinetic

### Car on flat curve

Friction produces centripetal acceleration

Tendency of car to move in straight line causes tires to pull against pavement as car turns. Newton's 3d law: pavement pulls in opposite direction of tires. Frictive force points toward center of curve; thus, the car turns.

static: part of the tire in contact with road is at rest for a short time, it is not sliding

kinetic: the part of the tire in contact with the road is sliding against the surface

Fnet<ma_c means that car will begin to slide

The car sliding is related to it's mass, square of velocity, and turning radius

Smaller turns require lower speed

### Car on banked curve

Horizontal component of normal force contributes to centripetal acceleration

With no friction, a car going too slowly will tend to slide inward while a car going to quickly will tend to slide outward.

### Ferris Wheel

circular motion is vertical

Heavier at bottom due to greater upward pressure from seat, much like elevator passenger

Lighter at top because rider weight only produces centripetal acceleration

### Motion on a slope (not in text)

The motion of an object on a slope must be broken into horizontal and vertical vectors.

X

Imagine the \ leg has an object at the center of the X. It is the slope on which the object lies. It represents the horizontal (x) component of the vectors. Kinetic force F_k lies uphill of this point.

The / leg represents the vertical (y) component of the vectors. F_n (normal force) lies uphill, w lies downhill.

Notice that if this is tilted such that the horizontal vector is level that it is the same as a force diagram for an object lying on a table.

Sketch a base level that the two legs lie upon. Through the center point, drop a line perpendicular to the base. Note that this forms two complementary right triangles. For the downslope triangle, theta represents the angle at the base. For the triangle under the uphill portion, theta represents the apex angle.

F_n (top of y vector)=w*cos(theta)

w (bottom of y vector)=cos(theta)

w (bottom of perpendicular line) = m*g

F_k (top of x vector)=Mu_k**Fn=Mu_k**w*cos(theta)

F_? (bottom of x vector)=w*sin(theta)

Mu_s=tan(theta)

a_x=F_net/m

F_net=F_?-F_k=w**sin(theta)-Mu_k**w*cos(theta)

### Kepler's Laws

1 Planets all move in elliptical orbits about a sun, which located at one focus of the ellipse

2 An imaginary line drawn from the sun to any planet moves through equal areas in equal intervals of time

3 If T is the amount of time of a period and r is the average radius of the orbit then T²/r³ is the same for all known planets

r³_earth**T²_mars=r³_mars**T²_earth

Period: time to complete a full orbit around the sun. On earth, a year.