Problem Set 3, Gravitation, Friction, Springs, Resistive Forces, and Circular Motion

Topics 2.6 through 2.10. Take a_g = −9.80665 ĵ m/s² and G = 6.674 × 10⁻¹¹ N·m²/kg². Out of 100 points.

The same six problems as the printed set. Beside every part there is a button that opens Socrates on that part alone. He will not give you the answer and he will not do the algebra: he asks you one question at a time until you get there yourself. Print the sheet for the parts that ask you to draw or plot.

Print the sheet

Problem 117 points

Treat the Earth as a uniform sphere of mass M = 5.972 × 10²⁴ kg and radius R = 6.371 × 10⁶ m. A narrow tunnel is imagined straight through the center from one side to the other, and a small object of mass m is released from rest at one end. (2.6)

(a)Find the magnitude of the gravitational field at the surface, and state what the shell theorem says about an object inside a thin spherical shell.4 pts

(b)At a distance r from the center, only part of the sphere pulls on the object. Write that partial mass in terms of M, R and r, and show that the force on the object is −GMmr/R³.4 pts

(c)That force has the form −kr. Find k for m = 1.00 kg and hence the period of the resulting oscillation, in seconds and in minutes. Sketch the force against r on the grid.4 pts

(d)Find the period of a satellite in a circular orbit skimming the surface, compare it with (c), and account for what you find.5 pts

Problem 217 points

A 4.00 kg block sits on a ramp inclined at 32.0°. A light string runs from the block, up the slope and over an ideal pulley at the top, to a block hanging freely. Between block and ramp the coefficient of static friction is 0.450 and the coefficient of kinetic friction is 0.310. (2.7)

(a)Draw the free-body diagram of the block on the ramp and find the largest static friction force the surface can supply.4 pts

(b)Find the range of hanging masses for which the system stays at rest.4 pts

(c)The hanging mass is 6.00 kg. Find the acceleration of each block and the tension in the string.4 pts

(d)Find the acceleration of the center of mass of the two-block system as a vector, and say why its magnitude is not equal to the magnitude of either block’s acceleration.5 pts

Problem 316 points

Two ideal springs have stiffnesses k₁ = 240 N/m and k₂ = 360 N/m. A 3.00 kg block hangs at rest from them. (2.8)

(a)The springs are side by side, both attached to the ceiling and both to the block. Find the equivalent stiffness and the stretch of each spring.4 pts

(b)The springs are instead joined end to end, the first to the ceiling and the second to the block. Find the equivalent stiffness, the stretch of each spring, and the total extension.4 pts

(c)Show that the series combination is always softer than its softer spring and the parallel combination always stiffer than its stiffer spring, and give the physical reason for each in one sentence.4 pts

(d)The 240 N/m spring is replaced by a nonideal one that exerts F_s = k₁ x + bx³ with b = 1.50 × 10³ N/m³, used alone. Find the stretch under the same block and say whether the spring is stiffer or softer than the ideal one at this extension.4 pts

Problem 417 points

An object of mass 0.150 kg falls from rest through air that exerts a resistive force F_r = −k v with k = 0.240 kg/s. (2.9)

(a)Write Newton’s second law for the object as a differential equation in v, and find the terminal speed from it without solving the equation.4 pts

(b)Separate the variables and integrate over stated limits to obtain v(t). Sketch it on the grid and mark the asymptote.4 pts

(c)Find the time to reach 90.0 percent of the terminal speed, and the speed at t = 1.00 s.4 pts

(d)The same object slides on frictionless ice at 8.00 m/s with the same resistive force and no driving force. Find v(t) and the total distance it travels, and explain how that distance can be finite when the object never formally stops.5 pts

Problem 516 points

A curve of radius 62.0 m is banked at 18.0°. The coefficient of static friction between the tires and the road is 0.380. (2.10)

(a)Draw the free-body diagram of a car on the curve and find the speed at which no friction at all is needed.4 pts

(b)Find the greatest speed at which the car can hold the curve without slipping.4 pts

(c)Find the least such speed, or show that there is none, and say what your answer means for a car stopped on the bank.4 pts

(d)Rain drops the coefficient to 0.150. Find the new greatest and least speeds, and say why the mass of the car appears in none of these answers.4 pts

Problem 617 points

A satellite is in a circular orbit of radius 4.22×10⁷ m about the center of the Earth, whose mass is 5.972×10²⁴ kg. (2.10)

(a)Find the period of the orbit, in seconds and in hours, and say what is special about the number you get.4 pts

(b)Find the orbital speed and the centripetal acceleration, then check the acceleration a second way.4 pts

(c)Use only the period and the radius to recover the mass of the Earth.4 pts

(d)A second satellite has a period exactly half the first. Find its orbital radius and speed, say which satellite is moving faster, and explain why that is the opposite of what pushing something harder would do.5 pts