, SOLUTIONS MANUAL
Introduction to Classical Mechanics
With Problems and Solutions
David Morin
Cambridge University Press
,TO THE INSTRUCTOR: I have tried to pay as much attention to detail in
these exercise solutions as I did in the problem solutions in the text. But
despite working through each solution numerous times during the various
stages of completion, there are bound to be errors. So please let me know if
anything looks amiss.
Also, to keep this pdf file from escaping to the web, PLEASE don’t distribute
it to anyone, with the exception of your teaching assistants. And please make
sure they also agree to this. Once this file gets free, there’s no going back.
In addition to any comments you have on these solutions, I welcome any
comments on the book in general. I hope you’re enjoying using it!
David Morin
(Version 2, April 2008)
c David Morin 2008
°
, Chapter 1
Strategies for solving
problems
1.8. Pendulum on the moon
The only way to get units of time from ℓ, g, and m is through the combination
p
ℓ/g. Therefore,
p
√
r
TM ℓ/gM gE
= p = =⇒ TM ≈ 6 TE ≈ 7.3 s. (1)
TE ℓ/gE gM
1.9. Escape velocity
(a) Using M = ρV , we have
r
2G · (4/3)πR3 ρ p
v= = (8/3)πGR2 ρ. (2)
R
√
(b) We see that v ∝ R ρ. Therefore,
√
vJ RJ ρJ 1
= √ = 11 · √ = 5.5. (3)
vE RE ρE 4
1.10. Downhill projectile
The angle β is some function of the form, β = f (θ, m, v0 , g). In terms of units, we
can write 1 = f (1, kg, m/s, m/s2 ). We can’t have any m dependence, because there
is nothing to cancel the kg. And we also can’t have any v0 or g dependence, because
they would have to appear in the ratio v0 /g to cancel the meters, but then seconds
would remain. Therefore, β can depend on at most θ. (And it clearly does depend
on θ, because β = 90◦ for θ = 0 or 90◦ , but β 6= 90◦ for θ 6= 0 or 90◦ .)
1.11. Waves on a string
The speed v is some function of the form, v = f (M, L, T ). In terms of units, we can
write m/s = f (kg, m, kg m/s2 ). We need
p to get rid of the kg’s, so we must use the
ratio T /M . We then quickly see that LT /M has the correct units of m/s. Note
p
that this can also be written as T /ρ, where ρ is the mass density per unit length.
1.12. Vibrating water drop
The frequency ν is some function of the form, ν = f (R, ρ, S). In terms of units, we
can write 1/s = f (m, kg/m3 , kg/s2 ). Wep
need to get rid of the kg’s, so we must use
the ratio S/ρ. We then quickly see that S/ρR3 has the correct units of 1/s. Note
p
that this can also be written as S/M , where M is the mass of the water drop.
1
Introduction to Classical Mechanics
With Problems and Solutions
David Morin
Cambridge University Press
,TO THE INSTRUCTOR: I have tried to pay as much attention to detail in
these exercise solutions as I did in the problem solutions in the text. But
despite working through each solution numerous times during the various
stages of completion, there are bound to be errors. So please let me know if
anything looks amiss.
Also, to keep this pdf file from escaping to the web, PLEASE don’t distribute
it to anyone, with the exception of your teaching assistants. And please make
sure they also agree to this. Once this file gets free, there’s no going back.
In addition to any comments you have on these solutions, I welcome any
comments on the book in general. I hope you’re enjoying using it!
David Morin
(Version 2, April 2008)
c David Morin 2008
°
, Chapter 1
Strategies for solving
problems
1.8. Pendulum on the moon
The only way to get units of time from ℓ, g, and m is through the combination
p
ℓ/g. Therefore,
p
√
r
TM ℓ/gM gE
= p = =⇒ TM ≈ 6 TE ≈ 7.3 s. (1)
TE ℓ/gE gM
1.9. Escape velocity
(a) Using M = ρV , we have
r
2G · (4/3)πR3 ρ p
v= = (8/3)πGR2 ρ. (2)
R
√
(b) We see that v ∝ R ρ. Therefore,
√
vJ RJ ρJ 1
= √ = 11 · √ = 5.5. (3)
vE RE ρE 4
1.10. Downhill projectile
The angle β is some function of the form, β = f (θ, m, v0 , g). In terms of units, we
can write 1 = f (1, kg, m/s, m/s2 ). We can’t have any m dependence, because there
is nothing to cancel the kg. And we also can’t have any v0 or g dependence, because
they would have to appear in the ratio v0 /g to cancel the meters, but then seconds
would remain. Therefore, β can depend on at most θ. (And it clearly does depend
on θ, because β = 90◦ for θ = 0 or 90◦ , but β 6= 90◦ for θ 6= 0 or 90◦ .)
1.11. Waves on a string
The speed v is some function of the form, v = f (M, L, T ). In terms of units, we can
write m/s = f (kg, m, kg m/s2 ). We need
p to get rid of the kg’s, so we must use the
ratio T /M . We then quickly see that LT /M has the correct units of m/s. Note
p
that this can also be written as T /ρ, where ρ is the mass density per unit length.
1.12. Vibrating water drop
The frequency ν is some function of the form, ν = f (R, ρ, S). In terms of units, we
can write 1/s = f (m, kg/m3 , kg/s2 ). Wep
need to get rid of the kg’s, so we must use
the ratio S/ρ. We then quickly see that S/ρR3 has the correct units of 1/s. Note
p
that this can also be written as S/M , where M is the mass of the water drop.
1
