A box is sliding with a speed of 4.50 m/s on a horizontal surface when, at point P, it encounters a rough section. On the rough section, the coefficient of friction is not constant but starts at .100 at P and increases linerly with distance past P, reaching a value of .600 at 12.5 m past point P. (a) Use the work energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid iff the friciton coefficient didn't increase, but instead had the constant value of .1?

Question
Other
A box is sliding with a speed of 4.50 m/s on a horizontal surface when, at point P, it encounters a rough section. On the rough section, the coefficient of friction is not constant but starts at .100 at P and increases linerly with distance past P, reaching a value of .600 at 12.5 m past point P. (a) Use the work energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid iff the friciton coefficient didn't increase, but instead had the constant value of .1?

2021-03-24
Let x be the distance past P.
$$\displaystyle\mu_{{{b}}}={0.100}+{A}{x}$$
When $$\displaystyle{x}={12.5}{m}\mu_{{{b}}}={0.600}$$
$$\displaystyle{A}={\frac{{{0.500}}}{{{12.5}{m}}}}=\frac{{0.0400}}{{m}}$$
a)
$$\displaystyle{W}=\triangle{K}{E}:{W}_{{{f}}}={K}{E}_{{{f}}}-{K}{E}_{{{i}}}$$
$$\displaystyle-\int\mu_{{{b}}}{m}{g}{\left.{d}{x}\right.}={0}-{\frac{{{1}}}{{{2}}}}{m}{{v}_{{{i}}}^{{{2}}}}$$
$$\displaystyle{g}{\int_{{{0}}}^{{{x}_{{{f}}}}}}{\left({0.100}+{A}{x}\right)}{\left.{d}{x}\right.}={\frac{{{1}}}{{{2}}}}{{v}_{{{i}}}^{{{2}}}}$$
$$\displaystyle{g}{\left[{\left({0.100}\right)}{x}_{{{f}}}+{A}{\frac{{{{x}_{{{f}}}^{{{2}}}}}}{{{2}}}}\right]}={\frac{{{1}}}{{{2}}}}{{v}_{{{i}}}^{{{2}}}}$$
$$\displaystyle{\left({9.80}\frac{{m}}{{s}^{{{2}}}}\right)}{\left[{\left({0.100}\right)}{x}_{{{f}}}+{\left(\frac{{0.0400}}{{m}}\right)}{\frac{{{{x}_{{{f}}}^{{{2}}}}}}{{{2}}}}\right]}={\frac{{{1}}}{{{2}}}}{\left({4.50}\frac{{m}}{{s}}\right)}^{{{2}}}$$
Now solve for $$\displaystyle{x}_{{{f}}}$$, We get the answer as $$\displaystyle{x}_{{{f}}}={5.11}{m}$$
b)
$$\displaystyle\mu_{{{k}}}={0.100}+{\left(-.-\frac{{40}}{{m}}\right)}{\left({5.11}{m}\right)}={0.304}$$
c)
$$\displaystyle{W}_{{{f}}}={K}{E}_{{{f}}}-{K}{E}_{{{i}}}$$
$$\displaystyle-\mu_{{{b}}}{m}{g}{x}={0}-{\frac{{{1}}}{{{2}}}}{m}{{v}_{{{1}}}^{{{2}}}}$$
$$\displaystyle{x}={\frac{{{{v}_{{{1}}}^{{{2}}}}}}{{{2}\mu_{{{b}}}{g}}}}={\frac{{{\left({4.50}\frac{{m}}{{s}}\right)}^{{{2}}}}}{{{2}{\left({0.100}\right)}{\left({9.8}\frac{{m}}{{s}^{{{2}}}}\right)}}}}={10.3}{m}$$

Relevant Questions

A 10 kg objectexperiences a horizontal force which causes it to accelerate at 5 $$\displaystyle\frac{{m}}{{s}^{{2}}}$$, moving it a distance of 20 m, horizontally.How much work is done by the force?
A ball is connected to a rope and swung around in uniform circular motion.The tension in the rope is measured at 10 N and the radius of thecircle is 1 m. How much work is done in one revolution around the circle?
A 10 kg weight issuspended in the air by a strong cable. How much work is done, perunit time, in suspending the weight?
A 5 kg block is moved up a 30 degree incline by a force of 50 N, parallel to the incline. The coefficient of kinetic friction between the block and the incline is .25. How much work is done by the 50 N force in moving the block a distance of 10 meters? What is the total workdone on the block over the same distance?
What is the kinetic energy of a 2 kg ball that travels a distance of 50 metersin 5 seconds?
A ball is thrown vertically with a velocity of 25 m/s. How high does it go? What is its velocity when it reaches a height of 25 m?
A ball with enough speed can complete a vertical loop. With what speed must the ballenter the loop to complete a 2 m loop? (Keep in mind that the velocity of the ball is not constant throughout the loop).
A 2.4-kg object is attached to a horizontal spring of forceconstant k=4.5 kN/m. The spring is stretched 10 cm fromequilibrium and released. Find (a) the frequency of themotion, (b) the period, (c) the amplitude, (d) the maximum speed,and (e) the maximum acceleration. (f) When does the objectfirst reach its equilibrium position? What is itsacceleration at this time?
Two identical blocks placed one on top of the other rest on africtionless horizontal air track. The lower block isattached to a spring of spring constant k= 600 N/m. Whendisplaced slightly from its equilibrium position, the systemoscillates with a frequency of 1.8 Hz. When the amplitude ofoscillation exceeds 5 cm, the upper block starts to slide relativeto the lower one. (a) What are the masses of the twoblocks? (b) What is the coefficient of static frictionbetween the two blocks?
A 15.0 kg block is dragged over a rough, horizontal surface by a70.0 N force acting at 20.0 degree angle above the horizontal. The block is displaced 5.0 m, and the coefficient of kinetic friction is 0.3. Find the work done on the block by ; a) the 70.0 N force,b) the normal force, and c) the gravitational force. d) what is the increase in the internal energy of the block-surface system due to friction? e) find the total change in the kinetic energy of the block.
Mitch throws a 100g lump of clay at a 500gtarger, which is at rest on a horizontal surface. After impact, thetarget, including the attached clay slides 2.1m before stopping. Ifthe coefficient of friction is .50, find the speed of the claybefore impact?
A wagon with two boxes of Gold, having total mass 300 kg, is cutloose from the hoses by an outlaw when the wagon is at rest 50m upa 6.0 degree slope. The outlaw plans to have the wagon roll downthe slope and across the level ground, and then fall into thecanyon where his confederates wait. But in a tree 40m from thecanyon edge wait the Lone Ranger (mass 75.0kg) and Tonto (mass60.0kg). They drop vertically into the wagon as it passes beneaththem. a) if they require 5.0 s to grab the gold and jump out, willthey make it before the wagon goes over the edge? b) When the twoheroes drop into the wagon, is the kinetic energy of the system ofthe heroes plus the wagon conserved? If not, does it increase ordecrease and by how much?
A 0.30 kg ladle sliding on a horizontal frictionless surface isattached to one end of a horizontal spring (k = 500 N/m) whoseother end is fixed. The ladle has a kinetic energy of 10 J as itpasses through its equilibrium position (the point at which thespring force is zero).
(a) At what rate is the spring doing work on the ladle as the ladlepasses through its equilibrium position?
(b) At what rate is the spring doing work on the ladle when thespring is compressed 0.10 m and the ladle is moving away from theequilibrium position?
An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airpline's engine is first started, it applies a constant torque of $$\displaystyle{1950}\ {N}\cdot{m}$$ to the propeller, which starts from rest.
a) What is the angular acceleration of the propeller? Model the propeller as a slender rod.
b) What is the propeller's angular speed after making 5.00 revolutions?
c) How much work is done by the engine during the first 5.00 revolutions?
e) What is the instantaneous power output of the motor at the instant that the propeller has turne through 5.00 revolutions?
A 75.0-kg man steps off a platform 3.10 m above the ground. Hekeeps his legs straight as he falls, but at the moment his feettouch the ground his knees begin to bend, and, treated as aparticle, he moves an additional 0.60 m before coming torest.
a) what is the speed at the instant his feet touch theground?
b) treating him as a particle, what is his acceleration(magnitude and direction) as he slows down, if the acceleration isassumed to be constant?
c) draw his free-body diagram (see section 4.6). in termsof forces on the diagram, what is the net force on him? usenewton's laws and the results of part (b) to calculate the averageforce his feet exert on the ground while he slows down. expressthis force in newtons and also as a multiple of his weight.
A 2.0-kg projectile is fired with initial velocity components $$\displaystyle{v}_{{{0}{x}}}={30}$$ m/s and $$\displaystyle{v}_{{{0}{y}}}={40}$$ m/s from a point on the earth's surface. Neglect any effects due to air resistance. What is the kinetic energy of the projectile when it reaches the highest point in its trajectory? How much work was done in firing the projectile?