# Q.) A student has six textbooks, each 4.0 cm thick and 30 N inweight. What is the minimum work the student would have to do toplace the books in a vertical stack, starting with all the books onthe surface of the table?

Question
Other
Q.) A student has six textbooks, each 4.0 cm thick and 30 N inweight. What is the minimum work the student would have to do toplace the books in a vertical stack, starting with all the books onthe surface of the table?

2021-05-19
Let mg be the weight of each book
let t be the thichness of each book.
The 1st book is aready on the table, so no work needs to be done.
The 2nd book has to be lifted through a height of t to beplaced on top of the 1st book.
The 3rd book has to be lifted through a height of 2t to beplaced on top of the 2nd book. etc.
The general case can be given as,
$$\displaystyle{W}_{{1}}={m}{g}\times{t}$$
$$\displaystyle{W}_{{2}}={m}{g}\times{2}{t}$$
$$\displaystyle{W}_{{3}}={m}{g}\times{3}{t}$$
$$\displaystyle{W}_{{{n}-{1}}}={m}{g}\times{\left({n}-{1}\right)}{t}$$
So, if you have n books, of thickness t, then the work done inlifting (n-1) of them on top of each other is given by,
$$\displaystyle{W}={m}{>}{\left({1}+{2}+{3}+\ldots+{\left({n}-{1}\right)}\right)}$$
You have a simple summation of the 1st (n-1) integers which is $$\displaystyle{\frac{{{1}}}{{{2}}}}{\left({n}-{1}\right)}{n}$$ so,
$$\displaystyle{W}={\frac{{{1}}}{{{2}}}}{m}{>}{\left({n}-{1}\right)}{n}$$
$$\displaystyle{m}{g}={30}{N}$$
$$\displaystyle{t}={0.04}{m}$$
$$\displaystyle{n}={6}$$
$$\displaystyle{W}={\frac{{{1}}}{{{2}}}}\times{30}\times{0.04}\times{5}\times{6}$$
$$\displaystyle{W}={15}\times{0.2}\times{6}$$
$$\displaystyle{W}={18}{J}$$

### Relevant Questions

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?
Water is being boiled in an open kettle that has a 0.500-cm-thick circular aluminum bottle with a radius of 12.0cm. If the water boils away at a rate of 0.500 kg/min,what is the temperature of the lower surface of the bottom of the kettle? Assume that the top surface of the bottom of the kettle is at $$\displaystyle{100}^{\circ}$$ C.
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?
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).
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
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?
A medical technician is trying to determine what percentage of apatient's artery is blocked by plaque. To do this, she measures theblood pressure just before the region of blockage and finds that itis $$\displaystyle{1.20}\times{10}^{{{4}}}{P}{a}$$, while in the region of blockage it is $$\displaystyle{1.15}\times{10}^{{{4}}}{P}{a}$$. Furthermore, she knows that blood flowingthrough the normal artery just before the point of blockage istraveling at 30.0 cm/s, and the specific gravity of this patient'sblood is 1.06. What percentage of the cross-sectional area of thepatient's artery is blocked by the plaque?
The student engineer of a campus radio station wishes to verify the effectivencess of the lightning rod on the antenna mast. The unknown resistance $$\displaystyle{R}_{{x}}$$ is between points C and E. Point E is a "true ground", but is inaccessible for direct measurement because the stratum in which it is located is several meters below Earth's surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance $$\displaystyle{R}_{{y}}$$. The procedure for finding the unknown resistance $$\displaystyle{R}_{{x}}$$ is as follows. Measure resistance $$\displaystyle{R}_{{1}}$$ between points A and B. Then connect A and B with a heavy conducting wire and measure resistance $$\displaystyle{R}_{{2}}$$ between points A and C.Derive a formula for $$\displaystyle{R}_{{x}}$$ in terms of the observable resistances $$\displaystyle{R}_{{1}}$$ and $$\displaystyle{R}_{{2}}$$. A satisfactory ground resistance would be $$\displaystyle{R}_{{x}}{<}{2.0}$$ Ohms. Is the grounding of the station adequate if measurments give $$\displaystyle{R}_{{1}}={13}{O}{h}{m}{s}$$ and R_2=6.0 Ohms?