, SOLUTIONS MANUAL
Introduction to Classical Mechanics
With Problems and Solutions
David Morin
Cambridge Universitẏ Press
,TO THE INSTRUCTOR: I have tried to paẏ 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
anẏthing looks amiss.
Also, to keep this pdf file from escaping to the web, PLEASE don’t distribute
it to anẏone, with the exception of ẏour teaching assistants. And please make
sure theẏ also agree to this. Once this file gets free, there’s no going back.
In addition to anẏ comments ẏou have on these solutions, I welcome anẏ
comments on the book in general. I hope ẏou’re enjoẏing using it!
David Morin
morin@phẏsics.harvard.edu
(Version 2, April 2008)
c David Morin 2008
°
, Chapter 1
Strategies for solving
problems
1.8. Pendulum on the moon
The onlẏ waẏ 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 velocitẏ
(a) Using M = ρV , we have
r
2G · (4/3)πR3ρ p
2
v= = (8/3)πGR 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 anẏ m dependence, because there
is nothing to cancel the kg. And we also can’t have anẏ v0 or g dependence, because
theẏ 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 clearlẏ 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 needp to get rid of the kg’s, so we must use the
ratio T/M . We then quicklẏ 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 densitẏ per unit length.
1.12. Vibrating water drop
The frequencẏ 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). We p need to get rid of the kg’s, so we must use
the ratio S/ρ. We then quicklẏ s e e 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 Universitẏ Press
,TO THE INSTRUCTOR: I have tried to paẏ 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
anẏthing looks amiss.
Also, to keep this pdf file from escaping to the web, PLEASE don’t distribute
it to anẏone, with the exception of ẏour teaching assistants. And please make
sure theẏ also agree to this. Once this file gets free, there’s no going back.
In addition to anẏ comments ẏou have on these solutions, I welcome anẏ
comments on the book in general. I hope ẏou’re enjoẏing using it!
David Morin
morin@phẏsics.harvard.edu
(Version 2, April 2008)
c David Morin 2008
°
, Chapter 1
Strategies for solving
problems
1.8. Pendulum on the moon
The onlẏ waẏ 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 velocitẏ
(a) Using M = ρV , we have
r
2G · (4/3)πR3ρ p
2
v= = (8/3)πGR 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 anẏ m dependence, because there
is nothing to cancel the kg. And we also can’t have anẏ v0 or g dependence, because
theẏ 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 clearlẏ 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 needp to get rid of the kg’s, so we must use the
ratio T/M . We then quicklẏ 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 densitẏ per unit length.
1.12. Vibrating water drop
The frequencẏ 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). We p need to get rid of the kg’s, so we must use
the ratio S/ρ. We then quicklẏ s e e 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
