Page 24 - [David_Morin]_Problems_and_Solutions_in_Introducto(OSN ASTRO_Neat

1.4. MULTIPLE-CHOICE QUESTIONS 15

1.9. The drag force F d on a sphere moving slowly through a viscous fluid depends on the
viscosity of the fluid η (with units kg/(m s)), the radius R, and the speed v. Which of the
following quantities is F d ?
2 2
2
(a) 6πηR/v (b) 6πη/Rv (c) 6πηRv (d) 6πηR v (e) 6πηR v
1.10. The drag force F d on a sphere moving quickly through a nonviscous fluid depends on the
density of the fluid ρ, the radius R, and the speed v. Which of the following quantities is
F d proportional to?
2 2
2
(a) ρv (b) ρRv (c) ρRv 2 (d) ρR v (e) ρR v
1.11. The Schwarzschild radius R S of a black hole depends on its mass m, the speed of light c,
3
2
and the gravitation constant G (with units m /(kg s )). Which of the following quantities
is R S ?
2G 2Gm 2Gm 2c 2 2c 3
(a) (b) (c) (d) (e)
mc 2 c 2 c 3 Gm Gm
B
In the remaining questions, don’t solve things from scratch. Just check special cases.
A
1.12. The plane in Fig. 1.2 is inclined at an angle θ, and two vectors are drawn. One vector
C D
is perpendicular to the plane, and its horizontal and vertical components are shown. The
other vector is horizontal, and its components parallel and perpendicular to the plane are
shown. Which of the following angles equal(s) θ? (Circle all that apply.) θ
(a) A (b) B (c) C (d) D
Figure 1.2
1.13. Two massless strings support a mass m as shown in Fig. 1.3. Which of the following
quantities is the tension (that is, force) T in each string?
mg mg sin θ mg cos θ mg mg
(a) (b) (c) (d) (e)
2 2 2 2 sin θ 2 cos θ
T T
θ θ
1.14. A block slides down a plane inclined at angle θ. What should the coefficient of kinetic
friction µ be so that the block slides with constant velocity?
(a) 1 (b) sin θ (c) cos θ (d) tan θ (e) cot θ Figure 1.3
1.15. Consider the “endcap” of the sphere shown in Fig. 1.4, obtained by slicing the sphere with
a vertical plane perpendicular to the plane of the paper. Which of the following expressions
is the volume of the cap? R
( ) θ
3
(a) πR 4/3 − (2/3) sin θ
( )
3
(b) πR (2/3) sin θ
( )
3
(c) πR 2/3 − (2/3) cos θ + sin θ
( )
3
3
(d) πR 2/3 + (1/3) cos θ − cos θ Figure 1.4
1.16. Consider the line described by ax + by + c = 0. Which of the following expressions is the
distance from this line to the point (x 0 , y 0 )?
bx 0 + ay 0 + c ax 0 + by 0 + c ax 0 + by 0 ax 0 + by 0 + c
(a) √ (b) √ (c) √ (d) √
2
2
2
2
2
a + b 2 a + b 2 a + b 2 a + b + c 2
1.17. A person throws a ball with a given speed v (at the optimal angle for the following task)
toward a wall of height h. Which of the following quantities is the maximum distance the
person can stand from the wall and still be able to throw the ball over the wall?
√
√
2
2
gh 2 v 2 v 4 v h v 2 2gh v /g
(a) (b) (c) (d) (e) 1 − (f)
2
v 2 g g h g g v 2 1 + 2gh/v 2