dynamics

When has velocity reached its max on a displacment time graph?

at point of inflection, second derivative = 0

equation for velocity not including time

how does accelerations direction affect particle

when acceleration is parallel to velocity- speed is changed not direction

when acceleration is perpendicular - speed is constant, direction is changed moves along curve

how is motion related in polar coordinates

theta is related to time, motion is not related to r, r and theta always perpendicular to eachother

what are unit vectors?

length 1 and dimensionless

what is r hat

what is the vector r

is a function of theta, associated with change in mag and not direction

what is theta hat

what is the velocity in polar coordinates

as magnitude and direction vary as r and theta move

what is the velocity in polar coordinates in r and theta form including components

radial component + tangential component

equation for angular velocity

forces in a circle

velocity needed to stay on top is larger or equal to sqrt (Rg)

what is r dot r hat

what are the radial and tangenital components in circular motion

a -radial , b-tangential

what is acceleration of an object moving in a circle

centripetal acceleration + Coriolis acceleration + linear acceleration

how to derive circular acceleration

this is because coordinate direction changes due to motion of the particle

what is newton’s first law/law of inertia

object continues moving if no external force has been applied to it, in the same inertial frame if object appears to not move as it has the same instantaneous velocity

what are the forces on a passenger sitting in the car and the car turns?

as car bends person continues travelling in a straight line (N1), if there is a large centripetal force acting on passenger, they move in curved path, the force is due to friction

if force is not large enough, person slides across seat

what is newton’s second law?

F=ma, vector, force is needed to change motion not maintain in,

what is newton’s second law applied to circular motion?

what is newton’s third law?

action and reaction are equal magnitudes, different in direction and act on different bodies, what does the work doesn’t always provide the force

what is the contact force

m2 is the object furthest from where the force was applied

what is friction

force which opposes relative motion of 2 surfaces in contact with each other

what is static friction

on a stationary object, when friction increases as F increases, so object stays in equilibrium

what is limiting friction

when pushing force becomes large enough and frction has reached its maximum so object begins to slide

what is limiting static friction

max friction experienced by stationary body

what is dynamic friction

friction experienced by a moving object, if object is pulled at a constant speed, force is less than force needed to start sliding

what is the friction coefficient

friction / normal

what is the friction coefficient on an inclined body

tan(theta)

why is friction proportional to normal force?

area of contact between objects is small, pressure is high at contact points and become flattened until weight is supported, strong intermolecular forces between them, moving objects need these forces to be broken, when broken they skim past each other ( less force needed for dynamic friction), when load increases, surface flattens further, greater force required

what is the speed of motion in a vertical circle

increases on way down, decreases on way up

what are the components of speed in a circle?

tangential = mg sin(theta)

radial = mg cos(theta) - T

what are the magnitudes on components in a circle

tangential = g sin(theta)

radial = -v²/R

what is the tension at any point in the circle

where theta = 0 in the south direction

what is the critical speed in a circle

must be at top otherwise string wouldn’t be taunt

what happens if weight is less than centripetal force

object leaves track and travels short parabola, loses KE, full loop is complete

what is work for objects in equilibrium

force x distance travelled x cos(theta)

what is the work done on an object when multiple forces act on an object

what is the work done when force varies?

what is the work done by a spring?

on a pendulum what is the work done

what is work done in 3d

force is a function of r

what is the change in work done

what is the work-energy theorem

work done by resultant force = change in kinetic energy of particle, when velocity is constant there is no work done

what is power

rate of doing work = force x velocity = work / time

equation of power in terms of work done

what is centre of mass

point that moves as if all mass was concentrated there and all external forces act on it

what is the equation to find centre of mass

for 3D replace r with vector r

what is rigid body and its centre of mass

system of packed particles

how to find acceleration of centre of mass

all forces acting on body are external, all internal forces cancel (N3)

what happens when an object moving in parabolic trajectory gets split in half

same external forces act on each, mass has decreases, trajectory continues, centre of mass of trajectory is the same as full objects trajectory

what is the kinetic energy of a system

KE of centre of mass + KE of particles relative to where they are to centre of mass

what is the reduced mass

what is the kinetic energy using reduced mass

what is gravitational potential energy

seperation between 2 objects causes attraction, shared between both objects, as GPE increases KE decreases

what is the GPE of an object

what is work done on a body when lifted

by gravity is negative, potential increases

what is elastic potential energy

energy associated with state of compression

what happens if elastic and gravitational force do work on a body

total mechanical energy is conserved, potential energy is sum of both

what are properties of a conservative force

work done is reversible, independent of path, expressed as difference between initial and final values of U

what is the elastic force

object collides with spring, speed is reducded until it is 0, block moves in opposite direction and gains speed,KE will be the same when it returns, mechanical energy is conserved

what is a non-conservative force

object has less KE than it started, friction does negative work in both directions, total mechanical not conserved

what is the total final mechanical energy

initial mechanical energy + work done by non-conservative forces = change in KE

what is the potential energy in 3D

only for conservative forces

what happens to potential if force is in same direction as displacement

decreases

what do conservative forces do to potential energy

push towards lower potential energy

elastic potential graph

total mechanical constant, forbidden region as v cannot be negative

where are the stable and unstable equilibria

what is momentum

vector, mass x velocity, is always conserved

what is the rate of change of momentum

proportional to resultant force, is in the same direction

what is the momentum of a centre of mass

what is impulse

rate of change of momentum

what is the average force of a system

what is an elastic collision

KE is consevered

what is a perfectly inelastic collision

KE is not conserved, objects stick together

what is a partially inelastic collision

KE is not conserved, objects move apart

what happens when one object is stationary when hit

moving object comes to rest, rest object has its velocity

what happens in an elastic collision

realtive velocity = -relative velocity after

how to find x-components of velocity

what happens in perfectly inelastic collisions if KE is greater after

potential energy was lost

what is the coefficient of restitution

what is a glancing collision

do not hit at centres, conservation of momentum applied in x and y directions

what is an impact parameter

initial distance between centres

what are the angles particles go when they collide

add up to 90degrees, velocities have dot product = 0

what are rotations of rigid bodies

angular displacment is the same for all, ds ( r x dtheta) is different

what is angular velocity

w

what is angular acceleration

what is the tangential velocity of rotating body

what is the tangential acceleration of rotating body

what is radial acceleration of rotating body

= centripetal accleration

how to find direction of vector

or use of right hand

what is KE of rotating body

what is moment of inertia

kg m², depends on how mass of object is distribute, further from centre = greater MoI

how to calculate moment of inertia

dm = density dV

dV = r dr dz dtheta

moment of inertia of objects

solid sphere- 2/5MR²

thin sphere - 2/3MR²

ring - MR²

solid circle - 1/2MR²

what is the torque

force x perpendicular distance (cross product), friction force x coefficient of friction

equation of torque from inertia

what is torque due to gravity

total gravitational force actin on CoM

what is parallel axis theorem

proof/example in notes

what is work done on rotating objects