Page 23 - [David_Morin]_Problems_and_Solutions_in_Introducto(OSN ASTRO_Neat

14 CHAPTER 1. PROBLEM-SOLVING STRATEGIES

Unless you have already completely solved a given problem,

Don’t just read through the solution!

If you read through a solution without first solving the problem, you will gain essentially nothing
from it.

1.4 Multiple-choice questions

As mentioned above: In the multiple-choice questions, be sure to fully commit to an answer
before checking to see if it is correct.
1.1. If the task of a given problem is to find a certain length, which of the following quantities
could be the answer? (The ℓ, v, a, t, and m in this and the following two questions are
given quantities with the dimensions of length, velocity, acceleration, time, and mass.)
√
2
(a) at (b) mvt (c) aℓ (d) v/t (e) v /a
1.2. If the task of a given problem is to find a certain time, which of the following quantities
could be the answer?
√ √
2
(a) a/t (b) mv/ℓ (c) v /a (d) ℓ/a (e) v/a
2
1.3. If the task of a given problem is to find a certain force (with units kg m/s ), which of the
following quantities could be the answer?
2
(a) mv 2 (b) mat (c) mv/t (d) mv/ℓ (e) v /ℓ
1.4. One mile per hour equals how many meters per second? (There are 1609 meters in a mile.)
(a) 0.04 (b) 0.45 (c) 1 (d) 2.2 (e) 27

In the following seven questions, don’t solve things from scratch. Just use dimensional analysis.
1.5. A block rests on an inclined plane with coefficient of friction µ (which is dimensionless).
Let θ max be the largest angle of inclination for which the block doesn’t slide down. Which
of the following is true?

(a) θ max is larger on the moon than on the earth.
(b) θ max is larger on the earth than on the moon.
(c) θ max is the same on the earth and the moon.
2
1.6. A mass m oscillates back and forth on a spring with spring constant k (with units kg/s ).
If the amplitude (the maximum displacement) is A, which of the following quantities is
the maximum speed the mass achieves as it passes through the equilibrium point?
√ √
k A k A 2 k A k A 2 √
(a) (b) (c) (d) (e) mk A 2
m m m m
3
1.7. A bucket of water with mass density ρ (with units kg/m ) has a small hole in it, at a depth
h below the surface. Assuming that the viscosity of the water is negligible, which of the
following quantities is the speed of the water as it exits the hole?
√ √ √ √ √
(a) 2gh (b) 2ρgh (c) 2g/h (d) 2h/g (e) 2gh/ρ
1.8. The increase in pressure ∆P (force per area) as you descend in a lake depends on your
depth h, the density of water ρ, and g. Which of the following quantities is ∆P?
2
2
(a) ρg/h (b) ρgh (c) ρ gh (d) ρg h (e) ρgh 3