Dynamics & Relativity 1

Question 1

for uniform motion r(t)=

Answer 1

r(t0)+v(to)(t-t0)+1/2A(t-t0)^2

Question 2

velocity

Answer 2

defined as rate of change of positions

v(t) = dr/dt = dxp/dt i + dyp/dt j + dzp/dt k

Question 3

acceleration

Answer 3

defined as the rate of change of velocity

a(t)=dv/dt=d^2r/dt^2 (same idea in vector components as velocity but with second derivative)

Question 4

simpler notation for time dependence

Answer 4

r0 = r(t0)
r = r(t)

hence r=ro+vo(t-t0)+1/2a(t-t0)^2

Question 5

motion in 1 dimension

Answer 5

pick just a single direction eg x

r=xi, v=vxi, a=axi

vx=dx/dt, ax=d&2x/dt^2 = constant

Question 6

derivation of vx=vx0+ax(t-t0) using integration

Answer 6

start with ax=dvx/dt and integration both sides wrt time

on lhs, a constant so get ax(t-t0)

rhs gives vx-vx0

put together a rearrange

Question 7

derivation of x-x0=vx0(t-t0)+ax/2(t-t0)^2

Answer 7

use vx=dx/dt

integrate both sides wrt time

insert previous equation for velocity and rearrange

Question 8

how to make x-x0=vx0(t-t0)+ax/2(t-t0)^2 more familiar?

Answer 8

setting initial position and time to zero gives

x=vx0t+1/2axt^2

Question 9

derivation of vx^2=vx0^2+2ax(x-x0)

Answer 9

combining previous two equations and eliminating (t-t0)

Question 10

derivation of x-x0=1/2(vx+vx0)(t-t0)

Answer 10

combining previous two equations and eliminating acceleration

Question 11

assumptions for free falling bodies

Answer 11

1. gravitational acceleration due to Earth’s gravity is constant
2. ignore gravity from everything but Earth
3. ignore rotation of Earth
4. Pretend Earth is flat
5. Ignore air resistance

Question 12

free falling body setup

Answer 12

ay=-g
vy=vy0-g(t-t0)
y-y0=vy0(t-t0)-1/2g(t-t0)^2

Question 13

free falling bodies - things to check

Answer 13

1. units
2. signs (heights +Ve or 0, object moving down so y-component of velocity is -ve
   3.magnitudes (timescale a few secodns, distance few tens of metres etc)

Question 14

motion in 2 dimensions

Answer 14

study the motion in each dimension seperately

Question 15

acceleration in x direction

Answer 15

0
hence vx=vx0

Question 16

separating velocity into components

Answer 16

vx0=v0costheta
vy0=v0sintheta

Question 17

how to find y as a function of x

Answer 17

take equation for y and use equation for x to eliminate t

Question 18

how far does an object travel (x direction)

Answer 18

set vertical position to zero and rearrange for Xr

Question 19

how high does object travel?

Answer 19

equations of motion for vy to find time when vy=0

or find dy/dx and set to 0 to find x at highest point. Use this to find y at highest point

or use symmetry if highest point is half of journey.

Question 20

maximum height only depends on

Answer 20

g and vy0

Question 21

relative motion

Answer 21

r p/a = r b/a + r p/b (similar for relative velocity if find derivative)

“the velocity of the point 𝑃
in coordinate system 𝐴 is equal to the
velocity of 𝑃 in coordinate system 𝐵 plus
the velocity of the origin of system 𝐵 as
measured in 𝐴”

Question 22

gradient of x-t plot

Answer 22

vx

Question 23

polar coordinates

Answer 23

r, theta

r=(x^2+y^2)^1/2
theta=tan^-1(y/x)

Question 24

w, angular spped

Answer 24

rate of change of theta

w=d theta/dt

(rad per second)

Question 25

time taken to complete revolution

Answer 25

T=2 pi/w

Question 26

r vector

Answer 26

<r cos theta, r sin theta>

Question 27

derivative of r vector

Answer 27

<-r sin theta, r cos theta>

Question 28

v is just r rotated by

Answer 28

90 degrees in the theta direction and multiplied by w

Question 29

for uniform circular motion, the velocity vector of a point is…

Answer 29

perpendicular to the position vector relative to the axis of rotation

Question 30

acceleration vector points…

Answer 30

in the opposite direction to the position vector relative to the axis of rotation

for uniform circular motion, a towards centre

Question 31

normal force

Answer 31

exerted on the box by the surface on which it is resting

always perpendicular to surface

Question 32

resultant force

Answer 32

sum of all forces acting on a body

Question 33

newton’s first law

Answer 33

a body acted on by no net force moves with constant velocity

Question 34

equilibrium

Answer 34

no net force

ie R=0

Question 35

Inertia

Answer 35

tendency of a body to remain at rest or keep moving if in motion

Question 36

inertial reference frame

Answer 36

if a reference frame accelerates, Newton’s first law does not hold.

inertial reference frame is one where Newton’s first law DOES hold

Question 37

Newton’s second law

Answer 37

if a net force acts on a body of mass m, the body will accelerate according to

ΣF=ma=R

Question 38

approach for forces questions

Answer 38

1. add up all forces to find ΣF=R (free body diagram useful)
2. use N2nd to find a
3. use a in eqns of motion

Question 39

if we plot the magnitude of frictional force applied to box against time

Answer 39

friction force matches pushing force, up to a maximum at which point the box begins to move and friction force reduces to a constant

Question 40

static friction

Answer 40

acts on surfaces not moving relative to one another

has maximum magnitude related to magnitude of the normal force due to contact between surfaces

Question 41

static friction equation

Answer 41

F < or = Fs

Fs=µsn

µs is the dimensionless coefficient of static friction

Question 42

kinetic friction

Answer 42

value proportional to the magnitude of the normal force acting on the object due to contact between the surfaces

Question 43

kinetic friction equation

Answer 43

Fk=µkn

µk is the dimensionless coefficient of kinetic friction

Question 44

coefficients of friction

Answer 44

depend on nature of surfaces

typically between 0 and 1 though can be >1

cannot be negative

µs>µk

Question 45

SHM

Answer 45

force acting on an object is proportional to its displacement but acts in the opposite direction