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

Topics 2.6 through 2.9. 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

Two uniform spheres sit at rest far from anything else. Sphere A has mass 500 kg and sphere B has mass 2000 kg, and their centers are 4.00 m apart. (2.6)

(a)Find the magnitude of the gravitational force each sphere exerts on the other, and say why the two forces are equal even though one sphere is four times the mass of the other.4 pts

(b)Find the magnitude of the gravitational field of sphere B at the center of sphere A, then use that field to recover your answer to (a).4 pts

(c)Somewhere on the line between the centers the two fields cancel. Plot the line on the grid to a stated scale and find that point, measured from the center of A.4 pts

(d)Show that your answer to (c) is unchanged if both masses are doubled, and say what happens to the point as sphere B is made heavier and heavier while sphere A is left alone.5 pts

Problem 217 points

A student of mass 65.0 kg stands on a bathroom scale in an elevator. The scale reads the magnitude of the normal force it exerts on the student. (2.6)

(a)Draw the free-body diagram of the student and find the scale reading while the elevator is at rest.4 pts

(b)Find the reading while the elevator accelerates upward at 2.40 m/s², and say whether the gravitational force on the student has changed.4 pts

(c)The cable fails and the elevator falls freely. Find the reading, and say in one sentence why the student is not weightless in the sense of having no gravitational force on them.4 pts

(d)Find the acceleration for which the scale would read exactly half the student’s weight, and state whether the elevator must be moving downward for that to happen.5 pts

Problem 316 points

A 12.0 kg crate sits on a level concrete floor. The coefficient of static friction between crate and floor is 0.520 and the coefficient of kinetic friction is 0.360. (2.7)

(a)Draw the free-body diagram of the crate while a horizontal push of 40.0 N is applied and the crate does not move, and state the magnitude and direction of the friction force.4 pts

(b)Find the smallest horizontal force that will start the crate moving.4 pts

(c)That same force is kept on the crate once it is sliding. Find the crate’s acceleration.4 pts

(d)The floor is slowly tilted with no push applied. Find the angle at which the crate begins to slide, and explain why the mass of the crate does not appear in your answer.4 pts

Problem 417 points

An ideal spring of stiffness 85.0 N/m has a relaxed length of 0.240 m. It is hung from a ceiling and a 1.50 kg block is attached to its lower end and released gently. (2.8)

(a)Draw the free-body diagram of the hanging block at rest, then find how far the spring has stretched and the total length of the spring.4 pts

(b)The block is pulled down a further 0.0600 m and held. Find the net force on it at that moment, with its direction.4 pts

(c)A classmate measures the spring’s stretch from the floor rather than from its relaxed length. Say what that does to the calculation and why the relaxed length is the only correct reference.4 pts

(d)The same spring and block are laid on a frictionless ramp at 25.0°, the spring fixed at the top and running along the surface. Find the stretch at rest and account exactly for the factor by which it differs from (a).5 pts

Problem 516 points

A 0.450 kg ball is whirled in a vertical circle of radius 0.850 m on the end of a light string. Later, a car rounds a curve of radius 55.0 m on a frictionless banked road. (2.9)

(a)Find the slowest speed the ball can have at the top of the circle and still travel the circle, and name the force that causes its centripetal acceleration at that instant.4 pts

(b)The ball passes the top at 4.00 m/s and later passes the bottom at 4.00 m/s. Find the string tension at each place, draw both free-body diagrams, and show that the difference is exactly twice the weight.4 pts

(c)The banked road is to be built so that a car needs no friction at all to hold the curve at 21.0 m/s. Find the banking angle.4 pts

(d)Say why the mass of the car does not appear in (c), and describe what happens to a car that takes the frictionless banked curve faster than 21.0 m/s.4 pts

Problem 617 points

A satellite moves in a circular orbit of radius 7.20 × 10⁶ m about the center of the Earth, whose mass is 5.972 × 10²⁴ kg. (2.9)

(a)Find the satellite’s orbital speed.4 pts

(b)Find the period of the orbit, in seconds and in hours.4 pts

(c)A second satellite orbits at twice that radius. Find the ratio of its period to the first satellite’s, without working out either period.4 pts

(d)Show that the orbital speed does not depend on the satellite’s own mass, then use only the period and radius from (a) and (b) to recover the mass of the Earth.5 pts