210
5 Gravitation
V =
4π R 3
3
(2)
∴ R =
3V
4π
1/3
(3)
∴ U = −
3
5
4π
3V
1/3
G M
2
(4)
P = −
∂U
∂ V
= −
1
5
4π
3
G M 2
V 4/3
∴ P ∝ V
−4/3
5.17 Consider a line element dx at A distance x from O, Fig. 5.13. The field point
P is at a distance R from the infinite line. Let PA = r . The x-component of
gravitational field at P due to this line element will get cancelled by a symmetric line element on the other side. However, the y-component will add up.
If λ is the linear mass density, the corresponding mass element is λdx
dE = dE y = −
Gλdx sin θ
r 2
(1)
Now, r
2
= x
2
+ R
2
(2)
x = R cot θ
(3)
∴ r
2
= R
2 cosec
2
θ
(4)
dx = R cosec
2
θ d θ
(5)
Using (4) and (5) in (1)
dE = −
Gλ
R
sin θ dθ
Fig. 5.13
5 Gravitation
V =
4π R 3
3
(2)
∴ R =
3V
4π
1/3
(3)
∴ U = −
3
5
4π
3V
1/3
G M
2
(4)
P = −
∂U
∂ V
= −
1
5
4π
3
G M 2
V 4/3
∴ P ∝ V
−4/3
5.17 Consider a line element dx at A distance x from O, Fig. 5.13. The field point
P is at a distance R from the infinite line. Let PA = r . The x-component of
gravitational field at P due to this line element will get cancelled by a symmetric line element on the other side. However, the y-component will add up.
If λ is the linear mass density, the corresponding mass element is λdx
dE = dE y = −
Gλdx sin θ
r 2
(1)
Now, r
2
= x
2
+ R
2
(2)
x = R cot θ
(3)
∴ r
2
= R
2 cosec
2
θ
(4)
dx = R cosec
2
θ d θ
(5)
Using (4) and (5) in (1)
dE = −
Gλ
R
sin θ dθ
Fig. 5.13