Classical Mechanics Flashcards
What is the differential equation for velocity?
v= dr/dt
And you can calculate r by integrating
What is the equation for linear momentum?
p=mv
What are Newton’s laws of motion?
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)
What are the SUVAT equations?
How can the velocity of someone falling be calculated in terms of time and constants? How does this relative to terminal velocity?
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
How can momentum calculations be carried out using vectors?
p=mv
momentum before= momentum directly after
If a vector, ensure components add to the same before and after
What is the equation for gravitational force between two objects?
F= -G (m1m2)/r2
where G is the gravitational constant
Normally gravitational force is negligible compared to electrostatic
What is the equation for work done?
dw= - F . dr
If in 2d, resolve forces into horizontal and vertical
We only care about vertical if F is horizontal, perpendicular distance
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?
=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
What is the equation for kinetic energy? What is the equation relating potential energy and force?
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
How do you calculate the mass of an ion in a mass spectrometer?
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
What is angular velocity and the equation for it?
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
How does angular velocity relate to linear velocity?
v= w x r
cross product
w and v are perpendicular as w out of the plane, r is the radius/distance
What is the equation of frequency from angular velocity? What about rotational time period?
freq(v)= w / 2π
as measured in radian
T=1/v = 2π/w
What is the formula for acceleration involving angular velocity? How has it been derived?
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
What is the formula for centripetal force and how is it derived?
f=ma
f=m x (v x w)
using v= w x r
f=mw2r
What is angular momentum?
J= r x p
= mvr
= mr2w
What is the moment of inertia?
I= mr2
and for a set of masses rotating about r, the moment of inertia is the sum of all the individual inertias
What is the reduced mass in general?
Motion of two masses replaced by the equivalent motion of one reduced mass about the centre of mass
How can you calculate the moment of inertia around BF3?
How has the reduced mass formula been calculated?
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
What is the equation for rotational kinetic energy? How has it been derived/
KE=1/2mv2
=1/2m(wr)2
=1/2 mr2w2
=1/2 I w2
1/2 x moment of inertia x angular velocity squared
What is the equation for force in a harmonic oscillator?
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
What is the solution for an equation relating displacement (x) and time for a harmonic oscillator?
x(t) = x0 cos (wt)
w= angular frequency here
What is the equation relating angular velocity and force constant? And frequency?
w= square root ( Kf/m)
v (freq)= 1/2π square root ( Kf/m)
What is the equation for potential energy in terms of Kf? How has it been derived?
F= -dV/dx
integrate f which is equal to -xKf
V(x)= 1/2 Kf x^2
What is the equation relating angular frequency and kf?
w= square root (Kf/reduced mass)
for diatomic molecules
