CC4A Translational Motion - Physics 1: Kinematics and Dynamics
A 10 kg mass is in free fall with no air resistance. In order to slow the mass at a rate equal to the magnitude of g, an upward force must be applied with magnitude: A) 0 N B) 10 N C) 100 N D) 200 N
***Remember that gravity is an acceleration. NOT a force. Answer is C is not correct because 100 N would only cancel our Fg and bring the object at a constant velocity. The question is asking you to slow the object's velocity. Therefore upward force must be greater than downward force so that acceleration vector can be directed upward and effectively slow velocity. Answer is D.
MCAT won't ask you to convert non-SI units to SI units. If it ever gives you non-SI units without providing you a conversion factor, it's probably a trick and info you don't need. Ex: A 200 kg car is going at a constant speed of 60 mph. Calculate the net force on the car.
0. **Only needed to know that the speed is CONSTANT **Constant speed = zero acceleration. Acceleration is required for any change in motion (whether it be speed or direction) **ZERO acceleration = zero NET force In this problem, there is no acceleration in the X or Y axis. Therefore net force on car is zero.
For the purposes of the MCAT, dimensions can be generalized into four types:
1D (One Dimension): Magnitude of length or distance (x or y line). => Ex: length of a string, distance traveled by a car. 2D: Length on a two-dimensional plane (x-y plane). => Ex: projectile motion of a ball. 3D: Length of distance in a three-dimensional space (xyz space). => Ex: inelastic collisions in a 3D space. 4D: Length in 3D space over a given time. => Time is the forth dimension here
The magnitude of the sum of two vectors must be:
=> no more than the sum of their magnitudes => no less than the difference of their magnitudes Ex: The sum of 7 m/s vector and 10 m/s vector can be no more than 17 m/s and no less than 3 m/s.
Newton's Second Law
A body of mass will accelerate when the vector sum of the forces results in some nonzero resultant force vector. **Note that the net force and acceleration vectors necessarily point in the same direction F(net)=ma **Force and acceleration are directly proportional: for a given mass, the greater the force, the greater the acceleration.
A projectile is fired from ground level with initial velocity of 50 m/s and an initial angle of elevation of 37°. A) The projectile's total time in flight B) Total horizontal distance traveled (sin 37°=0.6, cos 37°=0.8, g=10m/s)
A) => find the y-component of velocity vector: rsinθ Vy=30m/s => find t to max height: t=Vi/a t=3 s => find total time in flight: t to max height *2 t=6 s B) => find x-component of velocity vector: rcosθ Vx=40ms => multiply total time of flight by x component velocity: 40m/s(6s) x=240m
Air resistance (drag)
Air resistance is a FORCE and it results from an object's collisions with air molecules. The greater the number of molecular collisions that occur each second, the greater the effect of the drag. Shape, surface area, and velocity change the force of the air resistance that it experiences. (ex: reduced surface area also leads to fewer collisions and therefore less drag. as velocity increases, number of collisions also increase) ***The mass of an object does NOT affect the force of air resistance experienced by the object. But it does affect the acceleration that results from the air resistance because remember F=ma. ***If force is constant, acceleration and mass must be inversely proportional to one another. For this reason a golf ball and ping pong ball will experience similar forces of air resistance (bc of similar size and shape) but will experience different accelerations due to air resistance (F=ma). The direction of air resistance is opposite to the direction of motion. Thus, a projectile traveling up will have a downward acceleration greater than g because it will experience g and air resistance. **more massive objects are less affected by air resistance than are less massive objects. The reason why a ping pong will fall slower than a golf ball when both are dropped from same height, is because air resistance will slow down the ping pong more (F=ma).
Newton's First Law
Aka: Law of interia A body at rest or in motion with a constant velocity will remain that way unless a net force acts upon it. F(net)=ma
Acceleration
Any change in velocity is acceleration, whether it is a change in magnitude, direction, or both. Thus, a particle must accelerate in order to change a direction in its motion. **So an object can experience acceleration even though its speed hasn't changed. **Unlike displacement and velocity, velocity and acceleration do NOT have to be in the same direction. **Like displace and velocity, acceleration and NET force vector are always in the same direction (Newton's 2nd law)
What is the magnitude and direction of a vector with x component of 3 m/s and y component of 4 m/s?
Direction is θ = tan^-1 (y/x): θ = tan^-1(4/3) Magnitude is sqrt(x^2 + y^2) sqrt(4^2 + 3^2) = sqrt 25 = 5 (Note: this calculation is beyond the scope of the MCAT)
What is the difference between displacement and distance?
Displacement is a vector quantity that connects object's initial position and final position. It does not account for the actual pathway taken. Distance considers the pathway traveled and is a scalar quantity. EX: In one year the Earth travels roughly 940 million kilometers but its displacement is zero.
Units at molecular, atomic, or subatomic level (smaller scale)
Distance at atomic scale: => ångströms (1 Å = 10^-10 m) => nanometers (1 nm = 10^-9 m) Distance between two bp in DNA = 3.4 Å Distance in minor groove = 12 Å (1.2 nm) Distance in major groove = 22 Å (2.2 nm) Distance in 360° turn (minor + major groove) = 3.4 nm Width of DNA (from backbone to backbone) = 20 Å (2 nm) Energy at atomic scale: => electron-volts (1 eV = 1.6 * 10^-19 J) **which represents the amount of energy gained by an electron accelerating through a potential difference of one volt
1.4 Forces and Acceleration
Every change in velocity is motivated by a push or a pull --a force. This section we will examine how forces interact with one another and result in acceleration.
Hooke's Law
F = -k∆x **The force opposes direction of ∆x
Weight vs mass
F=mg F is the weight of the object m is its mass g is acceleration due to gravity
Centripetal force vs centripetal acceleration
Fc = mV^2/r centripetal acceleration (Fc/m): a = V^2/r *centripetal force can be caused by tension, gravity, electrostatic forces, or other forces
Gravitational Force
Fg = (Gm1m2)/r^2 **Fg is directly proportional to mass (if m1 is tripled, then Fg will triple) **Fg is inversely proportional to the square of the distance (if r is halved, then Fg will quadruple) G = 6.67 x 10^-11 m^3 kg^-1 s^-2 **Notice G is a very SMALL number (10^-11)
Inclined planes
Fg,||= mg sinθ Fg,⊥= mg cosθ Fg,||is the component of gravity parallel to the plane (oriented down the plane) Fg,⊥ is the component of gravity perpendicular to the plane (oriented into the plane) ***you can remember mgSinθ because the mass Slides down the incline ***on inclined planes, as θ increases, frictional force decreases.
How to find the Resultant (R) of V1 + V2 + V3
First need to break down vectors to x and y components Then get the Resultant x component and same for y. Then find the Resultant magnitude using pythagorean theorem. Find direction (θ) by taking the inverse tan of resultant x and y components: θ = arctan(y/x)
Adding vs subtracting vectors
Head-to-tail when adding vectors Head-to-head when subtracting vectors => can also do head-to-tail of the negative of a vector **The x-component of a resultant vector is simply the sum of all the x-components of the vectors being added. Same logic for y-component. **Adding vectors is commutative (order doesn't matter). Vector multiplication and subtraction, however, is not commutative and order matters!
Newton's 3rd law and how it relates to gravity
Newton's 3rd law states that the force of gravity on m1 from m2 is equal and opposite to the first of gravity on m2 from m1. That force equals: F=ma Although the force of gravity on myself and on Earth is equal in magnitude, Earth is much greater in mass. Therefore, I experience a much greater velocity towards the Earth, than the Earth does towards myself.
Will average speed always equal the magnitude of average velocity?
No. Instantaneous speed will always equal the magnitude of the object's instantaneous velocity (because instantaneous velocity is a measure of the average velocity as Δt approaches zero). BUT, average speed is not always magnitude of average velocity because avg speed is a measure of the total distance travelled over a given period of time. Whereas avg velocity is total displacement over period of time. So in the same unit of time, you can have a faster speed and travel more distance than a velocity representing the same start and end points. Ex: The average speed of the Earth over a year is about 30 kilometers per second. It's average velocity is 0.
A 5 kg block slides down a frictionless incline at 30º. Find the normal force and acceleration of the block.
Normal force = Fg,⊥= mg cosθ Normal force = 5kg(10m/ss)cos30º = 42.4N Acceleration = Fg,||/m = (mg sinθ)/m Acceleration = (5kg(10m/ss)sin30º)/5kg = 5m/ss
1.6 Motion with Constant Acceleration
Objects can undergo only two types of motion: => one which is constant (with no acceleration) => one that is changing (with acceleration)
Projectile Motion
Projectile motion is not linear motion, but its components are linear. First resolve projectile motion into its vertical and horizontal components. Then apply the linear motion equations.
How is gravity near Earth's surface compared to the gravity further away from Earth's surface? Why?
The force of gravity near Earth's surface is g = 10 N. => Further away from Earth's surface (height above the Earth) gravity decreases => Closer to Earth's center of mass (beneath the surface) gravity increases The magnitude of gravitational force between two objects is Fg = Gm1m2/r^2
A box rests on an incline. What happens to the forces on the box as the angle of inclination is increased?
The force parallel to the ramp (mg sinθ) increases as θ increases. The force perpendicular to the ramp (mg cosθ) decreases as θ increases.
A tennis ball and a feather dropped from the same height. Which will fall faster and why? (Assume that air resistance acts on both objects.)
The tennis ball has less surface area (less aerodynamic drag) And also the tennis ball has greater mass (larger gravitational force F=mg) so it can overcome its relatively less aerodynamic drag. For these reasons, the tennis ball will fall faster. **Air resistance is dependent on number of molecular collisions per second which depends on surface area, shape, and velocity.
A ping pong ball and a golf ball are both launched in a projectile at the same angle. In absence of air resistance, their trajectory is not the same. What could explain this?
Their initial velocities are not the same.
Uniformly Accelerated Motion and Linear Motion
These equations can only be used when an object is in constant acceleration and on a linear path. Use the equation that'll give you one unknown out of the following 4 variables: displacement, velocity, acceleration, and time. Displacement: => X - Xo = Vo*t + 1/2*A*t^2 **The half comes from the fact that area under v vs t graph is a trapezoid = 1/2(v1 + v2) * t and v2 is substituted by (v1 + at) Instantaneous Velocity: => V = Vo + At => V^2 = Vo^2 + 2a(X - Xo) Average velocity: V(avg) = 1/2(V+Vo)
Newton's Third Law
To every action, there is an equal but opposite reaction
To generate a _________ like work, we multiply vectors force and displacement using ___________. To generate a _________ like torque, we multiply vectors force and lever arm using ____________.
To generate a scaler quantity like work, we multiply vectors force and displacement using the DOT PRODUCT (A • B) A • B = |A| |B| cosθ To generate a vector quantity like torque, we multiply vectors force and lever arm using CROSS PRODUCT (A X B) A X B = |A| |B| sinθ **Use right-hand rule for cross products. Thumb is 1st vector, finger is 2nd, middle or palm is resultant (order matters). The resultant vector of a cross product (in this case, torque) will always be perpendicular to the plane created by the two vectors. **θ is the angle between the two vectors (for both dot and cross products) **Use the Pythagorean Theorem to get |A|and |B|. Ex: |A|= sqrt(Xa^2 + Ya^2)
1.7 Mechanical Equilibrium
Translational equilibrium means that the net forces acting on an object equals 0, therefore it has a constant velocity: both constant speed (could be zero or nonzero) and direction. Rotational equilibrium occurs when the vector sum of all the torques acting on an object is zero. This could mean either the object is rotating with a constant angular velocity or not rotating at all. **Rotation occurs around a fixed point called fulcrum **τ = r × F = rF sinθ
Friction
Unlike other types of forces like gravity and electromagnetic force, which can cause objects to speed up or slow down, friction forces almost always oppose an object's relative motion and causes it to slow down or become stationary. Two types of friction: => Static friction (fs): 0 ≤ fs ≤ μs*N => Kinetic friction (fk) fk = μk*N **the value of μs is always larger than the value of μk. Therefore the maximum value for static friction will always be greater than the constant value for kinetic friction: objects will stick until they start moving, and then will slide more easily over one another. => There's also drag (aka air resistance)
Kinematics (no displacement)
V = Vo + at initial velocity, final velocity, acceleration, time
Peak height of projectile Free falling objects
V = √(2gh) Projectile => h is maximum height projectile reaches when launched from the ground => V is the initial vertical velocity: Vo sinθ Free falling object => h is the height that a free falling object is dropped from => V is the final velocity of free falling object **this is neglecting air resistance: The path of a projectile is not influenced by its mass. Thus, a ping pong ball and golf ball will follow the same path IF THEIR INITIAL VELOCITIES ARE THE SAME
Kinematics (no time)
V^2 = Vo^2 + 2ax initial velocity, final velocity, acceleration, displacement
What is the difference between velocity and speed?
Velocity is a vector quantity and its magnitude is the change of displacement per second. Its direction is the same as displacement vector. Speed is a scaler quantity and measures the distance traveled in a given unit of time.
At what angle of launch will a projectile have the greatest horizontal displacement? What angle will result in the greatest vertical displacement?
Vertical DISTANCE: 90º horizontal displacement: 45º Horizontal displacement relies on both axis, so maximum horizontal displacement is achieved at 45º. Vertical displacement will always be zero if the object returns to the starting point. However, objects launched vertically will experience the greatest vertical distance.
A ball is thrown vertically up into the air from a window ledge 30 meters above ground with an initial velocity of 10 m/s. A) Find the velocity and position of the ball after 2 seconds B) Find the distance and the time the ball reaches maximum height.
We have 3 equations: ∆Y = Vt + 1/2at^2 (displacement) V = Vo + at (final velocity [in this case, when velocity = 0]) V^2 = Vo^2 + 2aY (max height when Y=0) A) Velocity at 2s: V = 10m/s - 9.8m/ss(2s) V= -9.6 m/s Position at 2s: Y=10m/s(2s) + 1/2(-9.8m/ss)(2s)^2 Y=.4m/s B) Time when ball reaches max height: 0=10m/s - 9.8m/ss(t) t=1s Distance of max height: Y=10m/s(1s)- 1/2(9.8m/ss)(1s)^2 Y= 5.1m -or- 0m/s= (10m/s)^2 - 2(9.8m/ss)(Y) -100m/s = -19.6m/ssY Y= 5.1m
1.3 Displacement and Velocity
We just covered the basic geometry that serves as the foundation of physics. Now we are going to examine the basic quantities that relate to kinematics: displacement, velocity, and acceleration.
When a vector is multiplied by a scaler, its _________ will change.
When a vector is multiplied by a scaler, its magnitude will change. Its direction will be either parallel or antiparallel to its original direction (depending if scaler is + or -). vector A multiplied by scaler n: B=nA **multiply vector A by the absolute value of n **If n is a positive number then B and A are in the same direction. If n is a negative number than B and A are in opposite direction.
If we know opposite and adjacent, we can find the hypotenuse using Pythagorean Theorem. Pythagorean Theorem will give us the MAGNITUDE only of the resultant vector.
X^2 + Y^2 = V^2 -or- V = sqrt(X^2 + Y^2) ***need inverse tan to get direction: θ=arctan(y/x)
The amount of time that an object takes to get to its max height is the same time it takes for the object to ____________________.
fall back to its starting height (displacement = 0). *this is assuming air resistance is negligible, which most of the time it is. **This can be used to find total time in flight (assuming end position is start position) by finding time it takes to reach max height (t=Vi/a) and multiplying that by 2. **By multiplying the time of flight by velocity in the x-direction, I can find the horizontal distance traveled.
Spring Constant
k = mg/∆x
See figure. Object A has twice the mass of object B (massA = 2 massB). The forces they exert on each other are _________. But their acceleration relative to a stationary point is __________.
m(a) = 2 m(b): mass of object a is twice that of object b F(a) = F(b): forces exerted on a by b is same as forces exerted on b by a 2 a(a) = a(b): acceleration of object b is twice that of object a. Adding the magnitudes of the objects' individual accelerations gives a value of 30 m/ss, the rate at which the objects are accelerating toward each other. Object B is accelerating relative to object A at 30 m/ss but it is accelerating relative to a stationary boundary A at only 20 m/ss.
MCAT will test on SI units
mass => kg time => s force => N (kgm/s^2) work and energy => Nm power => Nm/s
Common MCAT right triangles
opp,adj,hyp 3-4-5 triangle 5-12-13 triangle 1-1-sqrt2 => 45°, 45°, 90° 1-sqrt3-2 => 30°, 60°, 90°
Center of mass
x = (m1x1 + m2x2 + m3x3)/(m1 + m2 + m3) same for y and z axis. **You can only expect the center of gravity to be at the geometric center if the object is a homogenous body with symmetrical shape and uniform density. **Center of mass does not have to be on the object (ex: center of mass on a doughnut is the doughnut hole)
Kinematics (no final velocity):
x = Vot + (1/2)at^2 displacement, initial V, acceleration, time
Find the x- and y-components of the following vector v= 10 m/s θ = 30
x-component = 10(m/s) * cos30 = 10(m/s) (sqrt 3)/2 = 5 sqrt3 = 5.8 > x > 5.6 m/s y-component = 10(m/s) * sin30 = 5 m/s
If θ is the angle between V and it's x-component, what is y-component equal to?
y-component = rsinθ x-component = rcosθ **remember, cos related to x-component and sin related to y-component. As the angle increases from 0 to 90, x-component decreases and y-component increases. **component vectors are always at right angles to each other and their sum is equal to their resultant vector.
