Classical Mechanics

Question 1

What is the differential equation for velocity?

Answer 1

v= dr/dt
And you can calculate r by integrating

Question 2

What is the equation for linear momentum?

Answer 2

p=mv

Question 3

What are Newton’s laws of motion?

Answer 3

1st: A particle remains at rest or moves with constant speed in a straight line unless acted upon by a resultant force

2nd: F= dp/dt (p=momentum)
or F=ma

3rd: To every action there is always a opposite and equal reaction (force of A upon B is the same as B upon A but opposite direction)

Question 4

What are the SUVAT equations?

Answer 4

Question 5

How can the velocity of someone falling be calculated in terms of time and constants? How does this relative to terminal velocity?

Answer 5

F=Ma
F=F weight - F drag
F drag = -kv

Ma=Fw - Fd = mg - kv
a=dv/dt
m(dv/dt)= mg- kv
rearrange to separate the variables
Using limits of v(o) and v(t), and t=0 and t=t
End up with this form

Overtime, as a negative exponential multiplied by a constant, tends to terminal velocity

Question 6

How can momentum calculations be carried out using vectors?

Answer 6

p=mv
momentum before= momentum directly after
If a vector, ensure components add to the same before and after

Question 7

What is the equation for gravitational force between two objects?

Answer 7

F= -G (m1m2)/r2

where G is the gravitational constant

Normally gravitational force is negligible compared to electrostatic

Question 8

What is the equation for work done?

Answer 8

dw= - F . dr
If in 2d, resolve forces into horizontal and vertical
We only care about vertical if F is horizontal, perpendicular distance

Question 9

What is the equation for gpe? Where does it come from? How would you calculate the work done from separating charged particles from 0 to infinity?

Answer 9

=mgh, from the work equation, moving a mass for a certain height

W= -F.dr, using coulombs law for F, and integrating with respect to r from infinity and 0

Question 10

What is the equation for kinetic energy? What is the equation relating potential energy and force?

Answer 10

ke= 1/2 m v2

Fx = -dV/dx
where F is the force associated with the change in energy, V is the potential energy, and x the small change in position

As potential energy is energy possessed by an object due to its position

Question 11

How do you calculate the mass of an ion in a mass spectrometer?

Answer 11

electrical energy= charge x voltage (potential difference )
electrical energy=kinetic
using speed=distance/time, rearrange KE equation for mass
Mass will be in kg/mol, so multiple by avogadro’s and 1000 for g/mol, mass unit

Question 12

What is angular velocity and the equation for it?

Answer 12

When a particle is moving in a circle trajectory, with a constant radius , angular velocity is determined by the angle with the x axis

w(t)=dθ/dt

Question 13

How does angular velocity relate to linear velocity?

Answer 13

v= w x r
cross product
w and v are perpendicular as w out of the plane, r is the radius/distance

Question 13

What is the equation of frequency from angular velocity? What about rotational time period?

Answer 13

freq(v)= w / 2π
as measured in radian
T=1/v = 2π/w

Question 14

What is the formula for acceleration involving angular velocity? How has it been derived?

Answer 14

a=dv/dt
a=d/dt(v)=d/dt(w x r)
a= dr/dt x w
a= v x w

acceleration= linear velocity x angular velocity

Question 15

What is the formula for centripetal force and how is it derived?

Answer 15

f=ma
f=m x (v x w)
using v= w x r
f=mw2r

Question 16

What is angular momentum?

Answer 16

J= r x p
= mvr
= mr2w

Question 17

What is the moment of inertia?

Answer 17

I= mr2
and for a set of masses rotating about r, the moment of inertia is the sum of all the individual inertias

Question 18

What is the reduced mass in general?

Answer 18

Motion of two masses replaced by the equivalent motion of one reduced mass about the centre of mass

Question 19

How can you calculate the moment of inertia around BF3?

Answer 19

Question 20

How has the reduced mass formula been calculated?

Answer 20

Uses new centre=r1m1=r2m2
and r1 + r2= R (bond length)
and the moment of inertia equation

so I= reduced mass x bond length for a diatomic

Question 21

What is the equation for rotational kinetic energy? How has it been derived/

Answer 21

KE=1/2mv2
=1/2m(wr)2
=1/2 mr2w2
=1/2 I w2

1/2 x moment of inertia x angular velocity squared

Question 22

What is the equation for force in a harmonic oscillator?

Answer 22

Fx= -kfx
where kf is the force constant and x the displacement

= f=ma
ma=-kf x
so can become a differential equation replacing a with its derivative

Question 23

What is the solution for an equation relating displacement (x) and time for a harmonic oscillator?

Answer 23

x(t) = x0 cos (wt)

w= angular frequency here

Question 24

What is the equation relating angular velocity and force constant? And frequency?

Answer 24

w= square root ( Kf/m)

v (freq)= 1/2π square root ( Kf/m)

Question 25

What is the equation for potential energy in terms of Kf? How has it been derived?

Answer 25

F= -dV/dx
integrate f which is equal to -xKf
V(x)= 1/2 Kf x^2

Question 26

What is the equation relating angular frequency and kf?

Answer 26

w= square root (Kf/reduced mass)
for diatomic molecules