Transcript phy 2053 final review

PHY 2053 FINAL REVIEW
C. Buttery
Wednesday, Apr. 27th,
8:00-10:00
OVERVIEW
oVectors
oKinematics
oForces and Newton’s Laws
oCircular Motion
oWork and Energy
oImpulse
oMomentum
oRotational Kinematics
oRotational Dynamics (torque)
oHarmonic Motion
oQuestions and further clarifications
VECTORS
Vector – a quantity that has both
magnitude and direction
Scalar – has no direction
-speed
-velocity
-displacement
-acceleration
-force
-mass
-distance
KEEP IN MIND
oTrig functions
oSin, cos, tan
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
osin 𝜃 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ocos 𝜃 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
otan 𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
o Addition is tip-to-tail
oWhen in doubt, use components
oAll you need is 1 angle + 1 side or 2 sides to find anything else
PRACTICE PROBLEM 1
On a safari, a team of naturalists sets out toward a research station located 4.8 km
away in a direction 42° north of east. After traveling in a straight line for 2.4 km,
they stop and discover that they have been traveling 22° north of east, because their
guide misread his compass. What are the magnitude and direction (relative to due
east) of the displacement vector now required to bring the team to the research
station?
KINEMATICS
5 Kinematic Variables:
3 Equations:
1
2
oDisplacement, 𝑥 or 𝑦
o𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2
oFinal velocity, 𝑣𝑓 or 𝑣
o𝑣 = 𝑣0 + 𝑎𝑡
oInitial Velocity, 𝑣𝑜 or 𝑣𝑖
o𝑣 2 = 𝑣02 + 2𝑎(𝑥 − 𝑥0 )
oAcceleration, 𝑎
oTime, 𝑡
KEEP IN MIND
Write down everything you have
Separate it based on directions and cases
Maximum height? vfy = 0
𝑎𝑥 = 0
The final for one part of the problem could become the initial for another
Time depends on the y portion of the problem
Note your positive and negative directions
Draw everything!
PRACTICE PROBLEM 2
A jetliner, traveling northward, is landing with a speed of 69m/s. Once the jet
touches down, it has 750m of runway in which to reduce its speed to 6.1m/s.
Compute the average acceleration (magnitude and direction) of the plane during
landing.
PRACTICE PROBLEM 3
A tennis ball is thrown horizontally with and initial speed of 10 m/s. If the ball travels
40m horizontally, how far down did the tennis ball drop?
FORCES AND NEWTON’S LAWS
1st Law: An object continues in a state of rest or a state of motion at a constant speed
along a straight line unless compelled to change state by a net force
An object won’t want to change what it’s doing without a force
2nd Law: When a net external force acts on an object, the acceleration that results is
directly proportional to the net force and has a magnitude that is inversely
proportional to the mass. The direction of acceleration is the same as the direction of
the net force.
Basically, ∑𝐹 = 𝑚𝑎
3rd Law: Whenever one body exerts a force on a second body, the second body
exerts and oppositely directed force of equal magnitude on the first body
Equal and opposite reactions
TIPS FOR FORCE PROBLEMS
Draw a free body diagram. It always helps to see things
Write down literally everything
Equilibrium = no acceleration, therefore, ∑𝐹 = 0. Use this to your advantage
Mass is not always important. It may cancel in the end
Normal force is perpendicular to the surface. Not always mg!
Friction force is opposite to motion
Tension is along a rope
Gravitational force is straight down
PRACTICE PROBLEM 4
A 912-kg car is being driven down a straight, level road at a constant speed of 31.5
m/s. When the driver sees a police cruiser ahead, she removes her foot from the
accelerator. After 8.00 s, the speed of the car is 24.6 m/s, which is the posted speed
limit. What is the magnitude of the average net force acting on the car during the
8.00 s interval?
PRACTICE PROBLEM 5
A child decides to go sledding on a snowy day. The child is initially moving at 4.9
m/s, but eventually slows to a stop. Assuming that the coefficient of kinetic friction is
equal to 0.10, and the child’s weight is 50kg, what is the friction force? What is the
child’s acceleration?
PRACTICE PROBLEM 6
A worker stands still on a roof sloped at an angle of 36° above the horizontal. He is
prevented from slipping by a static frictional force of 390 N. Find the mass of the
worker.
CIRCULAR MOTION
Uniform Circular motion has a constant
speed, but a changing velocity.
Centripetal acceleration points inward,
towards the center of the circular motion.
HELPFUL EQUATIONS
𝑣=
2𝜋𝑟
𝑇
𝑣=
𝑎𝑐 =
(commonly used for circular orbits)
𝐺𝑀
𝑟
(also used for circular orbits. 𝐺 = 6.67408 × 10−11 m3 kg−1 s−2)
𝑣2
𝑟
𝐹𝑐 = 𝑚𝑎𝑐 =
𝑣2
𝑚
𝑟
𝑎𝑐 can also be denoted as 𝑎𝑟𝑎𝑑
* Unbanked curves depend on static friction force to provide the centripetal force,
whereas on a frictionless banked curve the centripetal force is provided by the
horizontal component of normal force (think back to your ramp problems!)
PRACTICE PROBLEM 7
A “swing” ride at a carnival consists of chairs
that are swung in a circle by 15.0-m cables
attached to a vertical rotating pole,
as the drawing shows. Suppose the total mass
of a chair and its occupant is 179 kg. What is the
Tension of the cable attached to the chair?
What is the speed of the chair?
PRACTICE PROBLEM 8
Civil Engineers generally bank curves on roads in such a manner that a car going
around the curve at the recommended speed does not have to rely on friction
between its tires and the road surface in order to round the curve. Suppose that the
radius of curvature of a given curve is 60m, and the recommended speed is 40km/hr.
At what angle should the curve be banked?
PRACTICE PROBLEM 9
Scientists determine the masses of planets by observing the effect of the
gravitational field of those planets on nearby objects - mainly upon their moons. By
measuring the orbital period and orbital radius of a moon about a planet, Newton's
laws of motion can be used to determine the mass of the planet. Phobos, a moon of
the planet Mars, was discovered in 1877. It's orbital radius is 9380 km and its orbital
period is 0.319 days (2.77 x 104 seconds). Determine the mass of Mars based on this
data.
WORK AND ENERGY
Work = Force * distance * cos𝜃
Work can also be measured as change in energy of a system.
- Potential Energy (mgh)
1
2
- Kinetic Energy ( m𝑣 2 )
CONSERVATION OF ENERGY
Energy is conserved if the system is only acted on by a conservative force
𝑊𝑛𝑐 = 0 = Δ𝐾𝐸 + Δ𝑃𝐸
Conservative:
Nonconservative:
•Gravity
•Static and Kinetic friction
•Elastic spring force
•Air resistance
•Electric force
•Tension
•Normal force
•Propulsion of a rocket
PRACTICE PROBLEM 10
A motorcyclist is trying to leap across a canyon by driving horizontally off a 70m high
cliff at 38.0 m/s to hit the 35m high cliff on the other side. Ignoring air resistance,
find the speed with which the cycle strikes the ground on the other side.
IMPULSE AND MOMENTUM
Impulse = Average force * time the force acts
𝐽 = 𝐹 ∗ Δ𝑡
Linear Momentum = mass * velocity
𝑝 = 𝑚𝑣
Impluse = change in linear momentum
𝐽 = Δ𝑝
*Note: If avg external force is 0, linear momentum is conserved.
COLLISIONS
Three major types to recognize:
1.) Elastic – Objects bounce off of each other
Kinetic energy and linear momentum conserved
2.) Inelastic – objects may bounce off, but not perfectly
Only linear momentum conserved, not kinetic energy
3.) Perfectly Inelastic – objects stick together after collision
Only linear momentum conserved, not kinetic energy
PRACTICE PROBLEM 11
Starting from rest, two skaters push off against each other on ice where friction is
negligible. One is a 54-kg woman and one is a 88-kg man. The woman moves away
with a speed of+2.5 m/s. Find the recoil velocity of the man.
ROTATIONAL KINEMATICS
Work exactly the same way as linear kinematics
Angular Displacement
𝜃
Angular Accelerations
α
𝜃=
Final Angular Velocity
𝜔 or 𝜔𝑓
𝜔2 = 𝜔02 + 2𝛼𝜃
Initial Angular Velocity
𝜔0 or 𝜔𝑖
Elapsed time
𝑡
1
𝛼𝑡 2
2
𝜔 = 𝜔0 + 𝛼𝑡
1
2
𝜔0 + 𝜔 𝑡
𝜃 = 𝜔0 𝑡 +
CONVERSIONS
𝑎𝑡 = 𝑟𝛼
𝑣𝑡 = 𝑟𝜔
𝑠 = 𝑟𝜃
𝑠 = arc length
All are tangental
PRACTICE PROBLEM 12
The wheels of a bicycle have an angular velocity of 20rad/s . Then, the brakes are
applied. In coming to rest, each wheel makes an angular displacement of 15.92rev.
How much time does it take for the bike to come to rest? What is the angular
acceleration (in rad/s 2 ) of the wheels?
ROTATIONAL DYNAMICS
Also known as torque
Most problems deal with relating the two equations for torque together.
𝜏 = 𝐼𝛼 = ∑𝐹𝑙𝑠𝑖𝑛𝜙
𝜙 is measured from the lever arm to the direction of the force
The maximum torque will be when 𝜙 = 90°, or when 𝑠𝑖𝑛𝜙 = 1
ROTATIONAL DYNAMICS
Rotational Kinetic Energy:
1
2
Angular Momentum:
𝐾𝐸𝑟 = 𝑅𝐸 = 𝐼𝜔2
𝐿 = 𝐼𝜔
𝜔 must be expressed in rad/s
Always conserved when no outside
torques act on the system
Side note: The torque is the rate of
change of the angular momentum
THINGS TO REMEMBER
If something is in equilibrium:




𝑎=0
∑𝐹 = 0
𝛼=0
∑𝜏 = 0
In general, 𝐼 = ∑𝑚𝑟 2 for the whole system
Formulas for specific moments of inertia will be provided
PRACTICE PROBLEM 13
A hiker, who weighs 985 N, is strolling through the woods and crosses a small
horizontal bridge. The bridge is uniform, weighs 3610 N, and rests on two concrete
supports, one at each end. He stops one-fifth of the way along the bridge. What is
the magnitude of the force that a concrete support exerts on the bridge at the near
end and at the far end?
PRACTICE PROBLEM 14
A bowling ball encounters a 0.760-m vertical rise on the way back to the ball rack,
as the drawing illustrates. Ignore frictional losses and assume that the mass of the ball
is distributed uniformly. The translational speed of the ball is 3.50 m/s at the bottom
of the rise. Find the translational speed at the top.
PRACTICE PROBLEM 15
Two disks are rotating about the same axis. Disk A has a moment of inertia of 3.4kg
𝑚2 and an angular velocity of 7.2 rad/s. Disk B is rotating with an angular velocity
of -9.8rad/s. The two disks are then linked together without the aid of any external
torques, so that they rotate as a single unit with an angular velocity of -2.4rad/s. The
axis of rotation for this unit is the same as that for the separate disks. What is the
moment of inertia of disk B?
HARMONIC MOTION
𝑘 is known as the spring constant (how stiff the spring is)
𝑎𝑝𝑝𝑙𝑖𝑒𝑑
𝐹𝑥
= 𝑘𝑥
This is how much force you apply to compress or stretch the string
The restoring force of an ideal spring (Hooke’s Law) is
𝐹 = −𝑘𝑥
HELPFUL EQUATIONS
Displacement: 𝑥 = 𝐴𝑐𝑜𝑠(𝜔𝑡)
Velocity:
𝑣 = −𝐴𝜔 sin 𝜔𝑡
Acceleration:
𝑎 = −𝐴𝜔2 cos(𝜔𝑡)
𝐴 = amplitutde = maximum displacement
𝜔 = angular speed = oscillations =
𝑘
𝑚
Maximum displacement/velocity/acceleration happens when the cos/sin equals 1
𝑣𝑚𝑎𝑥 = 𝐴𝜔
𝑎𝑚𝑎𝑥 = 𝐴𝜔2
MORE HELPFUL EQUATIONS
𝑇 = the time it takes to complete one cycle/revolution/etc
𝑓=
1
𝑇
𝜔 = 2𝜋𝑓 =
2𝜋
𝑇
Problems take a lot of relating different equations together.
PRACTICE PROBLEM 16
A vertical spring with a spring constant of 450 N/m is mounted on the floor. From
directly above the spring, which is unstrained, a 0.30-kg block is dropped from rest.
It collides with and sticks to the spring, which is compressed by 2.5 cm in bringing the
block to a momentary halt. Assuming air resistance is negligible, from what height (in
cm) above the compressed spring was the block dropped?
FINAL TIPS:
Don’t get overwhelmed!
Break things apart into smaller, more manageable chunks
For concepts: apply arbitrary numbers and see if that helps
Physics takes practice. Practice from the book, homework, etc
Things can be done in multiple ways
You’re all going to do super great! I believe in you! :D