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A.P. Physics 1 First Semester Review Sheet
Fall, Dr. Wicks Chapter 1: Introduction to Physics
- Review types of zeros and the rules for significant digits
- Review mass vs. weight, precision vs. accuracy, and dimensional analysis problem solving. Chapter 2: One-Dimensional Kinematics A. Velocity
- Equations for average velocity: f i ave f i
x^ x^ x
v
t t t
D^ -
D -
and
v ave = v f + vi
- In a position-versus-time graph for constant velocity, the slope of the line gives the average velocity. See Table 1.
- Instantaneous velocity can be determined from the slope of a line tangent to the curve at a particular point on a position-versus-time graph.
- Use vave =
Δ xTotal
Δ tTotal
to calculate the average velocity for an entire journey if given information about the various legs of the journey. B. Acceleration
- Equation for average acceleration: f i ave f i
v^ v^ v
a
t t t
D^ -
D -
- In a velocity-versus-time graph for constant acceleration, the slope of the line gives acceleration and the area under the line gives displacement. See Table 1.
- Acceleration due to gravity = g = 9.81 m/s^2. (Recall a = - g = - 9.81 m/s^2 )
Table 1: Graphing Changes in Position, Velocity, and Acceleration
Constant
Position
Constant
Velocity
Constant
Acceleration
Ball Thrown
Upward
Position
Versus
Time:
Velocity
Versus
Time:
Accelera
- tion
Versus
Time:
t
x
t
x
Slope = vave
t
x
t
x
t
v
t
v
t
v
Slope = aave
t
v
Slope = - 9.81 m/s^2
t
a
t
a
t
a
t
a
a = - 9.81m/s^2
Table 2: Comparing the Kinematic Equations Kinematic Equations Missing Variable
x = x o + vave t
a
v = v o + at D x
x = x o + v to + at
vfinal
2 2
v = vo + 2 a D x D t
Chapter 3: Vectors in Physics A. Vectors
- Vectors have both magnitude and direction whereas scalars have magnitude but no direction.
- Examples of vectors are position, displacement, velocity, linear acceleration, tangential acceleration, centripetal acceleration, applied force, weight, normal force, frictional force, tension, spring force, momentum, gravitational force, and electrostatic force.
- Vectors can be moved parallel to themselves in a diagram.
- Vectors can be added in any order. See Table 3 for vector addition.
- For vector r at angle q to the x-axis, the x- and y-components for r can be calculated from Δ x = r cos θ and Δ y = r sin θ.
- The magnitude of vector r is r = Δ x^2 + Δ y^2 and the direction angle for r relative to the
nearest x-axis is θ = tan
− 1 Δ y
Δ x
- To subtract a vector, add its opposite.
- Multiplying or dividing vectors by scalars results in vectors.
- In addition to adding vectors mathematically as shown in the table, vectors can be added graphically. Vectors can be drawn to scale and moved parallel to their original positions in a diagram so that they are all positioned head-to-tail. The length and direction angle for the resultant can be measured with a ruler and protractor, respectively. B. Relative Motion
- Relative motion problems are solved by a special type of vector addition.
- For example, the velocity of object 1 relative to object 3 is given by v
13 =^ v
12 +^ v
23 where object 2 can be anything.
- Subscripts on a velocity can be reversed by changing the vector’s direction: v
12 =^ − v
21
Chapter 4: Two-Dimensional Kinematics A. Projectile Motion
- See Table 4 to better understand how the projectile motion equations can be derived from the kinematic equations.
- The kinematic equations involve one-dimensional motion whereas the projectile motion equations involve two-dimensional motion. Two-dimensional motion means there is motion in both the horizontal and vertical directions.
- Recall that the equation for horizontal motion (ex. (^) D x = vx D t ) and the equations for vertical motion
(ex. vy f , = - g D t , 2
D y = - g D t ,^
2
vy f , = - 2 g D y ) are independent from each other.
- Recall that velocity is constant and acceleration is zero in the horizontal direction.
- Recall that acceleration is g = 9.81 m/s^2 in the vertical direction.
- When projectiles are launched at an angle, the range of the projectile is often calculated from
Δ x = ( vi cos θ )Δ t and its time of flight is often calculate from 2
( sin ) ( )
D y = v i q D - t g D t.
- Projectiles follow a parabolic pathway governed by 2 2
2 o
g
y h x
v
æ ö
= - ç ÷
è ø
Table 4: Relationship Between the Kinematic Equations and Projectile Motion Equations Kinematic Equations Missing Variable Projectile Motion, Zero Launch Angle Projectile Motion, General Launch Angle Assumptions made:
a = - g and vo y , = 0
Assumptions made:
a = - g , vo x , = vo cos q,
and (^) vo y (^) , = vo sin q
x = x o + vave t
a
D x = v tx where vx = const. D x = ( vo cos q) t where vx = const.
v = v o + at D x^ vy = - gt v y = vo sin q- gt
x = x o + v to + at
v final
D y = - gt
( sin )
D y = v o q t - gt
2 2
v = vo + 2 a D x D t^
2
vy = - 2 g D y
2 2 2 vy = vo sin q- 2 g D y
- For an object in free fall, the object stops accelerating when the force of air resistance, F
Air , equals
the weight, W
. The object has reached its maximum velocity, the terminal velocity.
- When a quarterback throws a football, the angle for a high, lob pass is related to the angle for a low, bullet pass. When both footballs are caught by a receiver standing in the same place, the sum of the launch angles is 90o.
- In distance contests for projectiles launched by cannons, catapults, trebuchets, and similar devices, projectiles achieve the farthest distance when launched at a 45o^ angle.
- The range of a projectile launched at initial velocity vo and angle q is 2
o sin 2
v
R
g
q
æ ö
= ç ÷
è ø
- The maximum height of a projectile above its launch site is 2 2 max
sin
y v^ o
g
q
Chapter 5: Newton’s Laws of Motion Table 5: Newton’s Laws of Motion Modern Statement for Law Translation Newton’s First Law: (Law of Inertia) Recall that mass is a measure of inertia. If the net force on an object is zero, its velocity is constant. An object at rest will remain at rest. An object in motion will remain in motion at constant velocity unless acted upon by an external force.
Newton’s Second Law: An object of mass m has an
acceleration a
given by the net
force F
∑ divided by^ m^. That
is a
F
m
F net = ma
Newton’s Third Law: Recall action-reaction pairs For every force that acts on an object, there is a reaction force acting on a different object that is equal in magnitude and opposite in direction. For every action, there is an equal but opposite reaction. A. Survey of Forces
- A force is a push or a pull. The unit of force is the Newton (N); 1 N = 1 kg-m/s 2
- See Newton’s laws of motion in Table 5. Common forces on a moving object include an applied force, a frictional force, a weight, and a normal force.
- Contact forces are action-reaction pairs of forces produced by physical contact of two objects. Review calculations regarding contact forces between two or more boxes.
- Field forces like gravitational forces, electrostatic forces, and magnetic forces do not require direct contact. They are studied in later chapters.
- Forces on objects are represented in free-body diagrams. They are drawn with the tails of the vectors originating at an object’s center of mass.
- Weight , W
, is the gravitational force exerted by Earth on an object whereas mass, m , is a measure of
the quantity of matter in an object ( W = mg ). Mass does not depend on gravity.
a , is the force felt from contact with the floor or a scale in an accelerating system. For example, the sensation of feeling heavier or lighter in an accelerating elevator.
, is perpendicular to the contact surface along which an object moves or is capable
of moving. Thus, for an object on a level surface, N
and W
are equal in size but opposite in direction.
However, for an object on a ramp, this statement is not true because N
is perpendicular to the surface of the ramp.
, is the force transmitted through a string. The tension is the same throughout the length of an ideal string.
Chapter 7: Work and Kinetic Energy A. Work
- A force exerted through a distance performs mechanical work.
- When force and distance are parallel, W = Fd with Joules (J) or Nm as the unit of work.
- When force and distance are at an angle, only the component of force in the direction of motion is used to compute the work: W = ( F cos q) d = Fd cosq
- Work is negative if the force opposes the motion ( q >90o). Also, 1 J = 1 Nm = 1 kg-m^2 /s^2.
- If more than one force does the work, then 1 n Total i i
W W
=
= å
- The work-kinetic energy theorem states that 2 2
W Total = D K = K f - Ki = mv f - mvi
- See Table 6 for more information about kinetic energy.
- In thermodynamics, (^) ( )
A F
W Fd Fd Ad P V
A A
æ ö æ ö
= = ç ÷ = ç ÷ = D
è ø è ø
for work done on or by a gas. Table 6 : Kinetic Energy Kinetic Energy Type Equation Comments Kinetic Energy as a Function of Motion:
K = mv
Used to represent kinetic energy in most conservation of mechanical energy problems. Kinetic Energy as a Function of Temperature:
ave ave
mv K kT
æ ö
ç ÷ =^ =
è ø
Kinetic theory relates the average kinetic energy of the molecules in a gas to the Kelvin temperature of the gas. B. Determining Work from a Plot of Force Versus Position
- In a plot of force versus position, work is equal to the area between the force curve and the displacement
on the x-axis. For example, work can be easily computed using W = Fd when rectangles are present
in the diagram.
- For the case of a spring force, the work to stretch or compress a distance x from equilibrium is
W = kx. On a plot of force versus position, work is the area of a triangle with base x
(displacement) and height kx (magnitude of force using Hooke’s Law, F = kx ).
C. Determining Work in a Block and Tackle Lab
- The experimental work done against gravity, WLoad , is the same as the theoretical work done by the spring scale, WScale.
- WOutput = WLoad = FdLoad = W
dLoad = mgdLoad where dLoad = distance the load is raised.
- WInput = WScale = FdScale where F = force read from the spring scale and (^) dScale = distance the scaled moved from its original position.
- Note that the force read from the scale is ½ of the weight when two strings are used for the pulley system, and the force read is ¼ of the weight when four strings are used.
D. Power
W
P
t
= or P = Fv with Watts (W) as the unit of Power.
- 1 W = 1 J/s and 746 W = 1 hp where hp is the abbreviation for horsepower. Chapter 8: Potential Energy and Conservation of Energy A. Conservative Forces Versus Nonconservative Forces
- Conservative Forces
- A conservative force does zero total work on any closed path. In addition, the work done by a conservative force in going from point A to point B is independent of the path from A to B. In other words, we can use the conservation of mechanical energy principle to solve complex problems because the problems only depend on the initial and final states of the system.
- In a conservative system, the total mechanical energy remains constant: (^) Ei = Ef. Since
E = U + K , it follows that U i + Ki = U f + Kf. See Table 6 for kinetic energy, K , and Table 7
for potential energy, U , for additional information.
- For a ball thrown upwards, describe the shape of the kinetic energy, potential energy, and total energy curves on a plot of energy versus time.
- Examples of conservative forces are gravity and springs.
- Nonconservative Forces
- The work done by a nonconservative force on a closed path is not zero. In addition, the work depends on the path going from point A to point B.
- In a nonconservative system, the total mechanical energy is not constant. The work done by a nonconservative force is equal to the change in the mechanical energy of a system:
W Nonconservative = Wnc = D E = E f - Ei.
- Examples of nonconservative forces include friction, air resistance, tension in ropes and cables, and forces exerted by muscles and motors. Table 7: Potential Energy Potential Energy Type Equation Comments Gravitational Potential Energy: U = mgh Good^ approximation^ for^ an object near sea level on the Earth’s surface. Gravitational Potential Energy Between Two Point Masses:
U G m m^1
r
= - where
G = 6.67 x 10-^11 Nm^2 /kg^2
= Universal Gravitation Constant Works well at any altitude or distance between objects in the
universe; recall that r is the
distance between the centers of the objects.
Elastic Potential Energy: 12
U = kx where k is the force
(spring) constant and x is the
distance the spring is stretched or compressed from equilibrium. Useful for springs, rubber bands, bungee cords, and other stretchable materials.
Chapter 10: Rotational Kinematics and Energy A. Rotational Motion
- Angular position, , in radians is given by where is arc length and is radius.
- Recall that θ( rad ) =
2 π radians
360 o
θ(deg).
- Counterclockwise (CCW) rotations are positive, and clockwise (CW) rotations are negative.
- In rotational motion, there are two types of speeds (angular speed and tangential speed) and three types of accelerations (angular acceleration, tangential acceleration, and centripetal acceleration). See Table 9 for a comparison.
- Since velocity is a vector, there are two ways that an acceleration can be produced: (1) changing the velocity’s magnitude and (2) changing the velocity’s direction. In centripetal acceleration, the velocity’s direction changes.
- When a person drives a car in a circle at constant speed, the car has a centripetal acceleration due to its changing direction, but it has no tangential acceleration due to its constant speed.
- The total acceleration of a rotating object is the vector sum of its tangential and centripetal accelerations. Table 9: Comparing Angular and Tangential Speed and Angular, Tangential, and Centripetal Acceleration Calculation Equations Units, Comments Angular Speed: (^) • radians/s
- Same value for horses A and B, side-by-side on a merry- go-round. Tangential Speed: (^) • m/s
- Different values for horses A and B, side-by-side on a merry-go-round. Angular Acceleration: •^ radians/s^2
- Same value for horses A and B, side-by-side on a merry- go-round. Tangential Acceleration: •^ m/s^2
- Different values for horses A and B, side-by-side on a merry-go-round. Centripetal Acceleration: •^ m/s^2
- is perpendicular to with directed toward the center of the circle and tangent to it.
D q
s
r
q
D
D = D s r
ave
t
q w
D
D
v t = r w ave
t
w a
D
D
a t = r a 2 t 2 c
v
a r
r
= = w
ac at
a c
a t
- Centripetal force, (^) FC , is a force that maintains circular motion: 2 t 2 C C
mv
F ma mr
r
= = = w
- The period, T , is the time required to complete one full rotation. If the angular speed is constant, then
T
p w
- The equations for rotational kinematics are the same as the equations for linear kinematics. See Table 10 for a comparison. Table 10: Kinematic equations for Rotational Motion Linear Equations Angular Equations x = x o + vave t θ = θ o + ω (^) avet v = v o + at ω = ω o + α t
x = x o + v to + at θ = θ o + ω ot +
α t
2
v = vo + a D x ω
o
- A comparison of linear and angular inertia, velocity, acceleration, Newton’s second law, work, kinetic energy, and momentum are presented in Table 11.
- The moment of inertia, , is the rotational analog to mass in linear systems. It depends on the shape or mass distribution of the object. In particular, an object with a large moment of inertia is difficult to start rotating and difficult to stop rotating. See Table 10-1 on p.298 for moments of inertia for uniform, rigid objects of various shapes and total mass.
- The greater the moment of inertia, the greater an object’s rotational kinetic energy.
- An object of radius , rolling without slipping, translates with linear speed and rotates with angular speed.
I
r v
v
r
w =
- Create an equation adding the torques together.
- If you are not sure what to do:
- Start by writing the forces again leaving some space between them.
- Multiply each force by an appropriate distance.
- Each force and distance pair will have the same subscript.
- Choose an axis of rotation in your diagram. It is helpful if your choice eliminates one of the two unknown forces. Then you can solve for the other force.
- Remember to enter correct signs in your torque equation. Consider whether each force will create a counterclockwise or clockwise rotation resulting in a positive or negative torque, respectively.
- Since the object is not moving, set the torque equation equal to zero. ( )
- Substitute numbers in your torque equation and solve for the unknown force.
- Substitute the force you determined in the last step into the force equation and solve for the other unknown force. C. Angular Momentum
- The rotational analog of momentum, , is angular momentum, in kg-m 2 /s, where = moment of inertia and angular velocity.
- If the net external torque acting on a system is zero, its angular momentum is conserved and. D. Simple Machines
- All machines are combinations or modifications of six fundamental types of machines called simple machines.
- Simple machines include the lever, inclined plane, wheel and axle, wedge, pulley(s), and screw.
- Mechanical advantage, , is defined as. It is a number describing how much force or distance is multiplied by a machine.
- Efficiency is a measure of how well a machine works, and is calculated using where is the work output and is the work input. Chapter 12: Gravity A. General Concepts About Gravity and Kepler’s Laws
- Newton’s Law of Universal Gravitation shows that the force of gravity between two point masses, m 1 and m 2 , separated by a distance r is 1 22
m m
F G
r
= where G is the universal gravitation constant,
G = 6.67x10-^11 Nm^2 /kg^2. Remember that r is the distance between the centers of the point masses.
- In Newton’s Law of Universal Gravitation, notice that the force of gravity decreases with distance, r , as (^2)
r
. This is referred to as an inverse square dependence.
- The superposition principle can be applied to gravitational force. If more than one mass exerts a gravitational force on a given object, the net force is simply the vector sum of each individual force. (The superposition principle is also used for electrostatic forces, electric fields, electric potentials, electric potential energies, and wave interference.)
å^ t^ =^0
p = mv L = I w I w =
L i = Lf
MA out^ in
in out
F d
MA
F d
% Efficiency
% Efficiency =
Wout
Win
⎟ (^100 )^ Wout^ Win
- Replacing the Earth with a point mass at its center, the acceleration due to gravity at the surface of the Earth is given by 2^ Earth Earth
GM
g
R
=. A similar equation is used to calculate the acceleration due to
gravity at the surface of other planets and moons in the Solar System.
- At some altitude, h , above the Earth, ( ) 2 Earth h Earth
M
g G
R h
can be used to calculate the acceleration due to gravity.
- In 1798, Henry Cavendish first determined the value of G, which allowed him to calculate the mass of the Earth using 2 Earth Earth
gR
M
G
- Using Tycho Brahe’s observations concerning the planets, Johannes Kepler formulated three laws for orbital motion as shown in Table 1 2. Newton later showed that each of Kepler’s laws follows as a direct consequence of the universal law of gravitation.
- As previously mentioned in the potential energy table, the gravitational potential energy, U , between two point masses m 1 and (^) m 2 separated by a distance r is 1 2
m m
U G
r
= -.^ This equation is used
in mechanical energy conservation problems for astronomical situations.
- Energy conservation considerations allow the escape speed to be calculated for an object launched from the surface of the Earth:
2 Earth
e Earth
GM
v
R
= = 11,200 m/s » 25,000 mi/h.
Table 1 2 : Kepler’s Laws of Orbital Motion Law Modern Statement for Law Alternate Description 1 st^ Law: Planets follow elliptical orbits, with the Sun at one focus of the ellipse. The paths of the planets are ellipses, with the center of the Sun at one focus. 2 nd^ Law: As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time. An imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals.
3 rd^ Law: The period, T , of a planet increases as its
mean distance from the Sun, r , raised to
the 3/2 power. That is,
S
T r
GM
æ (^) p ö
= ç ÷ =
ç ÷
è ø
(constant) (^) r 3/ 2 The square of a planet’s period, (^) T^2 , is proportional to the cube of its radius, 3
r.
That is, 2
S
T r
GM
p
= =(constant)
3
r
where (^) M (^) S = mass of the Sun.
