Cabrillo College
Physics 4A: Fall 2024
Due: Monday 10/7/24
Homework Set 4: Newtonian Gravitation & Centripetal Force
For all problems, begin with a pictorial representation of the problem. Then, starting from basic
physical rules & laws, mathematically derive the solution to the problem. This means you will
end up with an algebraic expression that is available for numerical analysis. I strongly encourage
you not to simply page through the textbook and pick an equation that solves your problem in one
step. It is essential for your problem-solving process that you understand completely how & why
the situation-specific equations are derived.
Be neat! Do not stint on space – please spread out!
Never insert numerical values until your algebra is completely worked through.
The single most important act in problem solving is…→ drawing a good picture!
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Key ideas in this chapter:
More rough estimation! It’s truly amazing what can be learned by patient and inquisitive observation.
We are continuing our study of Dynamics. Practice is crucial!
Drawing 2D force diagrams and setting up coordinates are the two biggest things you can master
to ensure success with these problems. Forces come in many functional forms but remember(!) …
they are only forces. Newton’s Laws are made to deal with forces.
Now is not the time to forget all your previous techniques!
Order of Magnitude Estimate / Trig Review
1. The world’s tallest building is in Dubai and towers at 2,717 feet tall. It has been claimed that the
tower may be seen from 60 miles away! Make a simple estimate and argue whether or not this could
be true.
Estimation
2. Estimate the total mass of Earth’s atmosphere by knowing only the gravitational field strength at
the surface, the size of Earth, and the standard air pressure at sea level (14.7 lbs per in2). Now find
the ratio of the mass of the oceans (refer to HW 2!) to the mass of the atmosphere.
Centripetal Forces
3. A pendulum (like fuzzy dice on a light string) hangs on your rearview mirror. As you round
a curve, you see an angle made by the string away from vertical. Find an expression for your
speed around the curve in terms of the curve’s radius R, and the angle (and g).
4. If a car goes through a curve too fast, it slides out of the curve. For a banked curve with friction,
the static frictional force acts on a fast car to oppose the tendency of the car to slide up the turn.
This frictional force is directed down the bank of the turn (the way water would drain). Consider a
circular curve of radius R and banking angle , with the maximum static coefficient of friction
between car and road of . Find an expression for the maximum speed the car can have going
around the turn (the car is on the verge of sliding out).
5. You are to design a circular exit ramp off a highway for traffic moving at 60 km/hr (17 m/s).
a. If the radius of the turn must be 30 m, what is the correct angle of banking so that no
skidding occurs even if the curve is covered in ice?
b. If the curve were not banked, what would be the minimum coefficient of static friction
required between the tires and the road that would keep traffic from skidding out when
traveling at 60 km/hr?
c. Suppose we now combine the two scenarios: the curve is banked as in (a) and the frictional
coefficient is as in (b). What is the maximum speed now possible without skidding? (Hint:
see your answer to #4.) (Answer: 65 m/s)
Gravitation
6. Left over from the big bang beginning of the universe, tiny black holes might still wander
through the universe. If one with a mass of 1011 kg (and a radius of only 10-16 m) reached
Earth, at what distance from your head would its gravitational pull on you match that of
Earth's?
7. The fastest possible rate of rotation for a planet is that where the gravitational force of the
mass at the equator provides the centripetal force needed for rotation. (Can you see why?)
3𝜋
Show that the shortest period of rotation is 𝑇 = √𝜌𝐺. Evaluate this for a density of 3 g/cm3.
No planet has ever been found to be spinning faster than this calculation.
8. Suppose we dropped Newton’s apple from 2 m above the surface of a neutron star (mass =
1.5x Sun’s mass, radius = 20 km). What is its approximate speed when hitting the surface?
(Answer: 1.4 x 106 m/s) What would be the approximate difference in g between the top and
bottom of the apple? (Hint: follow our in-class derivation for g.)
Drag Forces
9. Suppose that you achieve a final coasting speed of 20 km/hr while coasting down a hill
inclined at 200 on your bicycle. Assuming that air drag was the sole force limiting your final
speed, what speed would you then expect to coast if the hill was inclined at 350? (Answer: 26
km/hr)
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