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36 Cards in this Set
- Front
- Back
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Integrate gxh where g and h are vectors |
dg/dt x h + g x dh/dt |
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Drag force = |
0.5Cρav^2 |
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Terminal velocity = |
0.5CρAv^2 = mg V = sqrt ( 2mg/(CρA)) |
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Power with vectors |
P = F • V |
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Potential energy of a spring |
0.5kΔx^2 |
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How does F(x) relate to U(x)? |
F = - du/dx |
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Describe neutral equillibrium and give the sign/value of the second derivitive with x |
When the total mechanical energy is equal to the potential energy. Eg marble on a flat table. Second derivitive = 0 |
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Unstable equillibrium and give the sign/value of the second derivitive with x |
K.E = 0 but a small push will cause it to keep moving in the direction of the push. Eg a ball on a hill Second derivitive = -ve |
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Stable equillibrium and give the sign/value of the second derivitive with x |
K.E = 0 but a small push causes a restoring force. Eg a marble in a bowl, SHM Second derivitive = +ve |
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C.O.M = |
(1/M) Σ mi xi |
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Describe spatial symmetry and the conservation law associated: |
Physical laws are indipendent of position in the universe. Any origin may be chosen for the coordinate frame. Leads to the conservation of linear momentum. |
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Describe temporal symmetey and the conservation law associated |
Uniformity of time - any instant may be chosen as t=0. Leads to the conservation of energy. |
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Describe rotational symmetry and the conservation law associated |
Co-ordinate axes may be orientated in any way. This leads to the conservation of angular momentum. |
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What must be integrated to find the second rocket equation? |
dv = - Vrel 1/m dm |
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Impulse is the area under: |
F(t) vs t |
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Radial velocity = |
rω |
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radial acceleration = |
(v^2)/r = (rω)^2/r = rω^2 |
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Kinetic energy of rotation = |
K = 1/2 ι ω^2 |
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Moment of inertia = |
Σ m r^2 = r^2 dm = |
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Work done in angular displacement |
W = integral τ dθ |
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Power in angular displacement |
P = dW/dt = τ dθ/dt = τω |
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What is the parallel axis theorem |
Iparallel = Icom + h^2 M Where h = perpendicular distance between the parallel axes |
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Outline how to find the moment of inertia for a uniform rod |
Use length density u = M/L = dm/dx and I = integral x^2 dm to find the integral I = M/L integral x^2 dx and solve between xf and xi |
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Moment of inertia for a hoop |
I = integral R^2 dm = R^2 integral dm (sine R^2 is constant) = mR^2 |
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Moment of inertia for a uniform disc |
Ariel density = M/πR^2 = dm/2πrdr I = integral r^2 dm I = 2M/R^2 integral r^3 dr Solve for r=R and r=0 |
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Describe torque as a vector cross product |
τ = r x F |
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Magnitude of a torque = |
rFsin θ |
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Give the vector sum for angular momentum with and without mass as a term |
l = r x p = m(r x v) |
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Outline how to show N nd law in angular form |
Differentiate angular momentum with respect to t to find dl/dt = r x Fnet = τnet |
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Gravitatonal potential energy = |
-G Mm/r |
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Outline how to find the escape speed of an object |
If k = 0.5 mv^2 and u = - GMm/r At infinite distance u = 0 = k hence k + u = 0 . Substitute and solve for v |
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Outline the law of areas proof |
dA/dt = 0.5 r^2 dθ/dt = 0.5 r^2 ω Angular momentum l = mr^2 ω dA/dt = l/2m Hence if l is conserved then dA/dt = constant |
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Which two equations must be combined for the simple law of areas proof |
G Mm/r^2 = mrω^2 T = 2π/ω |
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The reduced mass of a system = |
m1m2 / (m1+m2) |
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Kinetic energy of a two body orbital system |
Ek = 0.5 μ R^2 ω^2 |
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Give the two equations for the barycentre law of periods |
T^2 = 4π^2r^3 / G(M+m) T^2 = [4π^2 (M+m)^2 / Gm^3] dm^3 |
