mechanics

Question 1

Hookes law equation

Answer 1

T = λx / l

Question 2

assumptions for hookes law

Answer 2

• the string/ spring is light
• The system has no friction
• The string / spring obeys hookes law

Question 3

hookes law on two springs:

Answer 3

1. draw a diagram for lengths, draw a diagram for tensions
2. find the forces + equate if theyre in equilibrium
3. apply hookes law to the tensions in the strings
4. sub these tensions into the equations made in step 2
5. solve

Question 4

work done by a variable force in a string/ spring

Answer 4

λx²/2l

Question 5

in perfectly elastic strings:

Answer 5

energy is conserved:

Final KE, GPE, EPE = Initial KE, GPE, EPE

Question 6

if a non conservative force does work on a string

Answer 6

work done = change in energy

Question 7

Solving vertical motion involving elastic forces using energy

Answer 7

1. calculate loss/ gain in GPE and KE

2. use conservation of energy to work out λ or x

Question 8

Solving vertical motion involving elastic forces using calculus

Answer 8

1. Use F = ma where a is a derivative
2. solve differential equation
3. input initial variables
4. input question conditions

Question 9

all quantities can be expressed in terms of:

Answer 9

Mass
Length
Time

Question 10

tangential speed, v =

Answer 10

rω

Question 11

radial acceleration is

Answer 11

acceleration towards the centre of the circle

Question 12

radial acceleration, a =

Answer 12

rω² = v²/r

Question 13

Force towards the centre of the circle, F =

Answer 13

mrω² = mv²/r

Question 14

Examples of centripetal force

Answer 14

• Tension in a string
• friction
• A reaction force

Question 15

In horizontal circles consider the forces:

Answer 15

• forces towards/ away from the centre of the circle

- forces perpendicular to the plane of motion

Question 16

working out speed in vertical circles

Answer 16

conservation of energy

GPE = KE

Question 17

tangential acceleration =

Answer 17

ra

a is the angular acceleration

Question 18

Constant angular acceleration with variable speed:

Answer 18

Suvat equations where 
ω = v
ωo = u
angular acceleration, a = a
t = t
θ = s

Question 19

If a particle just leaves its circular path then

Answer 19

R>0 T=0

Question 20

solving a circular motion problem

Answer 20

1. use forces (F=ma)
2. energy analysis
3. sub equations together and solve

Question 21

impulse =

Answer 21

Change in momentum = force * time

Ft = m(v-u)

Question 22

∫Fdt =

Answer 22

impulse

Question 23

∫Fds =

Answer 23

work done

Question 24

work done vector form

Answer 24

W = F.d

Question 25

power vector form

Answer 25

P = F.v

Question 26

Kinetic energy vector form

Answer 26

KE = ½mv.v

Question 27

alternate form for acceleration

Answer 27

dv/dt d²x/dt² vdv/dx

Question 28

a force may be dependent on:

Answer 28

- time F= f(t) - displacement F = f(x) - velocity F = f(v)

Question 29

including the effects of air resistance

Answer 29

1. F = ma Where F is the sum of forces and a is a derivative 2. separate variables 3. integrate and add arbitrary constant 4. use initial conditions to find constant 5. input conditions given in question

Question 30

When no forces acting except friction then

Answer 30

KE lost = work done against friction

Question 31

resultant acceleration in a circle =

Answer 31

√(radial acceleration²+tangential acceleration²)

Question 32

work energy pricipal

Answer 32

work done by forces = change in KE/GPE

Question 33

coefficient of restitution, e =

Answer 33

speed of separation / speed f approach

Question 34

if e = 0

Answer 34

the two velocities after impact are the same (the particles stick together), inelastic

Question 35

If e = 1

Answer 35

the loss in KE is zero, elastic

Question 36

Component of velocity perpendicular to the line of centres

Answer 36

unaffected by collision, momentum is conserved

Question 37

Component of velocity parallel to the line of centres

Answer 37

Newtons law of restitution, momentum is conserved

Question 38

When an object hits a plane there is no impulse...

Answer 38

parallel to the plane

Question 39

Component of velocity parallel to plane

Answer 39

unaffected by collision, momentum is conserved

Question 40

Component of velocity perpendicular to plane

Answer 40

Newtons law of restitution, momentum is conserved

Question 41

newtons law of restitution

Answer 41

(Ua - Ub ) e = Vb - Va

Question 42

Impulse = | integral

Answer 42

∫Fdt

Question 43

Work done = | integral

Answer 43

∫Fdx

Question 44

Using the impulse/ work done integrals

Answer 44

find the integral and set in equal to mv - mu or Pt

Question 45

component of velocity that changes wall vs two spheres

Answer 45

- wall: perpendicular component changes | - spheres: component parallel to line of centres changes

Question 46

stiffness =

Answer 46

λ/l

Question 47

x̄ =

Answer 47

(x1m1 + x2m2 + ... + xnmn)/ (m1 + m2 + ... + mn)

Question 48

ȳ =

Answer 48

(y1m1 + y2m2 + ... + ynmn)/ (m1 + m2 + ... + mn)

Question 49

finding CoM of compsite shaped laminas

Answer 49

in a table put each shapes area, CoM x co-ordinate, CoM y co-ordinate. Consider x or y co-ordinates i.e. x*A ... =x̄A

Question 50

If a hanging object is in equlibrium then...

Answer 50

the forces weight and tension act along the same line and there is no resultant moment. The CoM lies directly below the point of suspension.

Question 51

volume of revolution about x-axis

Answer 51

V = ∫πy²dx

Question 52

volume of revolution about y-axis

Answer 52

V = ∫πx²dy

Question 53

Vx̄ =

Answer 53

∫πy²xdx

Question 54

Vȳ =

Answer 54

∫πx²ydy

Question 55

Finding the CoM of a solid of revolution about the x-axis:

Answer 55

``` V = ∫πy²dx Vx̄ = ∫πy²xdx ```

Question 56

Finding the CoM of a solid of revolution about the y-axis:

Answer 56

``` V = ∫πx²dy Vȳ = ∫πx²ydy ```

Question 57

Finding the angle between the point of suspension and the vertical axes

Answer 57

- CoM directly below point of suspension so find x̄ and ȳ | - tanθ = ȳ/x̄

Question 58

CoM of a lamina using strips parallel to y-axis

Answer 58

``` Ax̄ = ∫xydx Aȳ = ∫½y²dx ```

Question 59

CoM of a lamina using strips parallel to the x-axis

Answer 59

``` Ax̄ = ∫½x²dy Aȳ = ∫xydy ```

Question 60

if an object is about to slide

Answer 60

F = μR | friction is limiting

Question 61

when the particle is at rest does friction act

Answer 61

no, there is no tendancy to slide

Question 62

an object will slide if

Answer 62

the line of action sits within the base

Question 63

an object will topple if

Answer 63

the line of action lies outside of the base

Question 64

volume of a circle

Answer 64

4/3πr^3

Question 65

volume of cone

Answer 65

h/3πr^2

Question 66

a in circular motion suvat

Answer 66

angular acceleration