Chapter 7

In one day, a \(75-\mathrm{kg}\) mountain climber ascends from the 1500 -m level on a vertical cliff to the top at 2400 \(\mathrm{m}\) . The next day, she descends from the top to the base of the cliff, which is at an elevation of 1350 \(\mathrm{m}\) . What is her change in gravitational potential energy (a) on the first day and \((b)\) on the second day?

A \(5.00-\mathrm{kg}\) sack of flour is lifted vertically at a constant speed of 3.50 \(\mathrm{m} / \mathrm{s}\) through a height of 15.0 \(\mathrm{m}\) . (a) How great a force is required? (b) How much work is done on the sack by the lifting force? What becomes of this work?

A \(120-\mathrm{kg}\) mail bag hangs by a vertical rope 3.5 \(\mathrm{m}\) long. \(\mathrm{A}\) postal worker then displaces the bag to a position 2.0 \(\mathrm{m}\) sideways from its original position, always keeping the rope taut. (a) What horizontal force is necessary to hold the bag in the new position? (b) As the bag is moved to this position, how much work is done (i) by the rope and (ii) by the worker?

A 72.0 -kg swimmer jumps into the old swimming hole from a diving board 3.25 \(\mathrm{m}\) above the water. Use energy conservation to find his speed just he hits the water (a) if he just holds his nose and drops in, (b) if he bravely jumps straight up (but just beyond the board) at 2.50 \(\mathrm{m} / \mathrm{s}\) , and (c) if he manages to jump downward at 2.50 \(\mathrm{m} / \mathrm{s} .\)

A baseball is thrown from the roof of a 22.0 -m-tall building with an initial velocity of magnitude 12.0 \(\mathrm{m} / \mathrm{s}\) and directed at an angle of \(53.1^{\circ}\) above the horizontal. (a) What is the speed of the ball just before it strikes the ground? Use energy methods and ignore air resistance. (b) What is the answer for part (a) if the initial velocity is at an angle of \(53.1^{\circ}\) below the horizontal? (c) If the effects of air resistance are included, will part (a) or (b) give the higher speed?

A crate of mass \(M\) starts from rest at the top of a frictionless ramp inclined at an angle \(\alpha\) above the horizontal. Find its speed at the bottom of the ramp, a distance \(d\) from where it started. Do this in two ways: (a) Take the level at which the potential energy is zero to be at the bottom of the ramp with \(y\) positive upward. (b) Take the zero level for potential energy to be at the top of the ramp with y positive upward. (c) Why did the normal force not enter into your solution?

An empty crate is given an initial push down a ramp, starting it with a speed \(v_{0},\) and reaches the bottom with speed \(v\) and kinetic energy \(K .\) Some books are now placed in the crate, so that the total mass is quadrupled. The coefficient of kinetic friction is constant and air resistance is negligible. Starting again with \(v_{0}\) at the top of the ramp, what are the speed and kinetic energy at the bottom? Explain the reasoning behind your answers.

A small rock with mass 0.20 \(\mathrm{kg}\) is released from rest at point \(A,\) which is at the top edge of a large, hemispherical bowl with radius \(R=0.50 \mathrm{m}\) (Fig. 7.25\()\) . Assume that the size of the rock is small compared to \(R,\) so that the rock can be treated as a particle, and assume that the rock slides rather than rolls. The work done by friction on the rock when it moves from point \(A\) to point \(B\) at the bottom of the bowl has magnitude 0.22 \(\mathrm{J}\) . (a) Between points \(A\) and \(B,\) how much work is done on the rock by (i) the normal force and (ii) gravity? (b) What is the speed of the rock as it reaches point \(B ?\) (c) Of the three forces acting on the rock as it slides down the bowl, which (if any) are constant and which are not? Explain. (d) Just as the rock reaches point \(B\) , what is the normal force on it due to the bottom of the bowl?

A stone of mass \(m\) is thrown upward at an angle \(\theta\) above the horizontal and feels no appreciable air resistance. Use conservation of energy to show that at its highest point, it is a distance \(v_{0}^{2}\left(\sin ^{2} \theta\right) / 2 g\) above the point where it was launched. (Hint: \(v_{0}^{2}=v_{\mathrm{ax}}^{2}+v_{\mathrm{oy}}^{2} . )\)

You are testing a new amusement park roller coaster with an empty car with mass 120 \(\mathrm{kg}\) . One part of the track is a vertical loop with radius 12.0 \(\mathrm{m}\) . At the bottom of the loop (point \(A\) ) the car has speed \(25.0 \mathrm{m} / \mathrm{s},\) and at the top of the loop (point \(B )\) it has speed 8.0 \(\mathrm{m} / \mathrm{s}\) . As the car rolls from point \(A\) to point \(B,\) how much work is done by friction?

Tarzan and Jane. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 \(\mathrm{m}\) that makes an angle of \(45^{\circ}\) with the vertical, steps off his tree limb, and swings down and then up to Jane's open arms. When he arrives, his vine makes an angle of \(30^{\circ}\) with the vertical. Determine, whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan's speed just before he reaches Jane. You can can can can ignore air resistance and the mass of the vine.

A 10.0 -kg microwave oven is pushed 8.00 \(\mathrm{m}\) up the sloping surface of a loading ramp inclined at an angle of \(36.9^{\circ}\) above the horizontal, by a constant force \(\vec{F}\) with a magnitude 110 \(\mathrm{N}\) and acting parallel to the ramp. The coefficient of kinetic friction between the oven and the ramp is 0.250 . (a) What is the work done on the oven by the force \(\vec{F} ?\) (b) What is the work done on the oven by the friction force? (c) Compute the increase in potential energy for the oven. (d) Use your answers to parts (a), \((b),\) and (c) to calculate the increase in the oven's kinetic energy. \((e)\) Use \(\Sigma \overrightarrow{\boldsymbol{F}}=m \overrightarrow{\boldsymbol{a}}\) to calculate the acceleration of the oven. Assuming that the oven is initially at rest, use the acceleration to calculate the oven's speed after traveling 8.00 \(\mathrm{m}\) . From this, compute the increase in the oven's kinetic energy, and compare it to the answer you got in part (d).

Pendulum. A small rock with mass 0.12 \(\mathrm{kg}\) is fastened to a massless string with length 0.80 \(\mathrm{m}\) to form a pendulum. The pendulum is swinging so as to make a maximum angle of \(45^{\circ}\) with the vertical. Air resistance is negligible. (a) What is the speed of the rock when the string passes through the vertical position? \((b)\) What is the tension in the string when it makes an angle of \(45^{\circ}\) with the vertical? (c) What is the tension in the string as it passes through the vertical?

A force of 800 \(\mathrm{N}\) stretches a certain spring a distance of 0.200 \(\mathrm{m} .\) (a) What is the potential energy of the spring when it is stretched 0.200 \(\mathrm{m} ?(\mathrm{b})\) What is its potential energy when it is compressed 5.00 \(\mathrm{cm} ?\)

An ideal spring of negligible mass is 12.00 \(\mathrm{cm}\) long when nothing is attached to it. When you hang a \(3.15-\mathrm{kg}\) weight from it, you measure its length to be 13.40 \(\mathrm{cm}\) . If you wanted to store 10.0 \(\mathrm{J}\) of potential energy in this spring, what would be its total length? Assume that it continues to obey Hooke's law.

A spring stores potential energy \(U_{0}\) when it is compressed a distance \(x_{0}\) from its uncompressed length. (a) In terms of \(U_{0},\) how much energy does it store when it is compressed (i) twice as much and (ii) half as much? (b) In terms of \(x_{0},\) how much must it be compressed from its uncompressed length to store (i) twice as much energy and (ii) half as much energy?

A slingshot will shoot a \(10-8\) pebble 220 \(\mathrm{m}\) straight up. (a) How much potential energy is stored in the slingshot's rubber band? (b) With the same potential energy stored in the rubber band, how high can the slingshot shoot a \(25-g\) pebble? (c) What physical effects did you ignore in solving this problem?

A spring of negligible mass has force constant \(k=\) 1600 \(\mathrm{N} / \mathrm{m}\) . (a) How far must the spring be compressed for 3.20 \(\mathrm{J}\) of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then drop a \(1.20-\mathrm{kg}\) book onto it from a height of 0.80 \(\mathrm{m}\) above the top of the spring. Find the maximum distance the spring will be compressed.

A 1.20 \(\mathrm{kg}\) piece of cheese is placed on a vertical spring of negligible mass and force constant \(k=1800 \mathrm{N} / \mathrm{m}\) that is compressed 15.0 \(\mathrm{cm}\) . When the spring is released, how high does the cheese rise from this initial position? (The cheese and the spring are not attached.)

A 2.50 -kg mass is pushed against a horizontal spring of force constant 25.0 \(\mathrm{N} / \mathrm{cm}\) on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 J of potential energy in it, the mass is suddenly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest acceleration of the mass, and when does it occur?

You are asked to design a spring that will give a \(1160-\mathrm{kg}\) satellite a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of 5.00 \(\mathrm{g}\) . The spring's mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will be negligible. (a) What must the force constant of the spring be? (b) What distance must the spring be compressed?

A \(75-\mathrm{kg}\) roofer climbs a vertical 7.0 -m ladder to the flat roof of a house. He then walks 12 \(\mathrm{m}\) on the roof, climbs down another vertical \(7.0-\mathrm{m}\) ladder, and finally walks on the ground back to his starting point. How much work is done on him by gravity (a) as he climbs up; \((b)\) as he climbs down; (c) as he walks on the roof and on the ground? (d) What is the total work done on him by gravity during this round trip? On the basis of your answer to part (d), would you say that gravity is a conservative or nonconservative force? Explain.

A \(10.0-\mathrm{kg}\) box is pulled by a horizontal wire in a circle on a rough horizontal surface for which the coefficient of kinetic friction is 0.250 . Calculate the work done by friction during one complete circular trip if the radius is (a) 2.00 \(\mathrm{m}\) and (b) 4.00 \(\mathrm{m}\) . (c) On the basis of the results you just obtained, would you say that friction is a conservative or nonconservative force? Explain.

In an experiment, one of the forces exerted on a proton is \(\overrightarrow{\boldsymbol{F}}=-\alpha x^{2} \hat{i},\) where \(\alpha=12 \mathrm{N} / \mathrm{m}^{2} .\) (a) How much work does \(\overrightarrow{\boldsymbol{F}}\) do when the proton moves along the straight- line path from the point \((0.10 \mathrm{m}, 0)\) to the point \((0.10 \mathrm{m}, 0.40 \mathrm{m}) ?(\mathrm{b})\) Along the straightline path from the point \((0.10 \mathrm{m}, 0)\) to the point \((0.30 \mathrm{m}, 0) ?\) (c) Along the straight-line path from the point \((0.30 \mathrm{m}, 0)\) to the point \((0.10 \mathrm{m}, 0) ?(\mathrm{d})\) Is the force \(\overrightarrow{\boldsymbol{F}}\) conservative? Explain. If \(\overrightarrow{\boldsymbol{F}}\) is conservative, what is the potential energy function for it? Let \(U=0\) when \(x=0\) .

A 0.60\(\cdot \mathrm{kg}\) book slides on a horizontal table. The kinetic friction force on the book has magnitude 1.2 \(\mathrm{N}\) . (a) How much work is done on the book by friction during a displacement of 3.0 \(\mathrm{m}\) to the left? (b) The book now slides 3.0 \(\mathrm{m}\) to the right, returning to its starting point. During this second 3.0 \(\mathrm{m}\) displacement, how much work is done on the book by friction? (c) What is the total work done on the book by friction during the complete round trip? (d) On the basis of your answer to part (c), would you say that the friction force is conservative or nonconservative? Explain.

You and three friends stand at the corners of a square whose sides are 8.0 \(\mathrm{m}\) long in the middle of the gym floor, as shown in Fig. \(7.26 .\) You take your physics book and push it from one person to the other. The book has a mass of 1.5 \(\mathrm{kg}\) , and the coefficient of kinetic friction between the book and the floor is \(\mu_{\mathrm{x}}=0.25 .\) (a) The book slides from you to Beth and then from Beth to Carlos, along the lines connecting these people. What is the work done by friction during this displacement? (b) You slide the book from you to Carlos along the diagonal of the square. What is the work done by friction during this displacement? (c) You slide the book to Kim who then slides it back to you. What is the total work done by friction during this motion of the book? (d) Is the friction force on the book conservative or nonconservative? Explain.

A block with mass \(m\) is attached to an ideal spring that has force constant \(k .\) (a) The block moves from \(x_{1}\) to \(x_{2},\) where \(x_{2}>x_{1} .\) How much work does the spring force do during this displacement? (b) The block moves from \(x_{1}\) to \(x_{2}\) and then from \(x_{2}\) to \(x_{1}\) . How much work does the spring force do during the displacement from \(x_{2}\) to \(x_{1} ?\) What is the total work done by the spring during the entire \(x_{1} \rightarrow x_{2} \rightarrow x_{1}\) displacement? Explain why you got the answer you did. (c) The block moves from \(x_{1}\) to \(x_{3},\) where \(x_{3}>x_{2}\) . How much work does the spring force do during this displacement? The block then moves from \(x_{3}\) to \(x_{2}\) . How much work does the spring force do during this displacement? What is the total work done by the spring force during the \(x_{1} \rightarrow x_{3} \rightarrow x_{2}\) displacement? Compare your answer to the answer in part (a), where the starting and ending points are the same but the path is different.

The potential energy of a pair of hydrogen atoms separated by a large distance \(x\) is given by \(U(x)=-C_{6} / x^{6},\) where \(C_{6}\) is a positive constant. What is the force that one atom exerts on the other? Is this force attractive or repulsive?

A force parallel to the \(x\) -axis acts on a particle moving along the \(x\) -axis. This force produces potential energy \(U(x)\) given by \(U(x)=\alpha x^{4},\) where \(\alpha=1.20 \mathrm{J} / \mathrm{m}^{4} .\) What is the force (magnitude and direction) when the particle is at \(x=-0.800 \mathrm{m} ?\)

Gravity in One Dimension. Two point masses, \(m_{1}\) and \(m_{2}\) , lie on the \(x\) -axis, with \(m_{1}\) held in place at the origin and \(m_{2}\) at position \(x\) and free to move. The gravitational potential energy of these masses is found to be \(U(x)=-G m_{1} m_{2} / x,\) where \(G\) is a constant (called the gravitational constant). You'll learn more about gravitation in Chapter 12 . Find the \(x\) -component of the force acting on \(m_{2}\) due to \(m_{1} .\) Is this force attractive or repulsive? How do you know?

Gravity in Two Dimensions. Two point masses, \(m_{1}\) and \(m_{2},\) lie in the \(x y\) -plane, with \(m_{1}\) held in place at the origin and \(m_{2}\) free to move a distance \(r\) away at a point \(P\) having coordinates \(x\) and \(y\) (Fig. 7.27\()\) . The gravitational potential energy of these masses is found to be \(U(r)=-G m_{1} m_{2} / r,\) where \(G\) is the gravitational constant. (a) Show that the components of the force on \(m_{2}\) due to \(m_{1}\) are $$ F_{x}=-\frac{G m_{1} m_{2} x}{\left(x^{2}+y^{2}\right)^{3 / 2}} \quad \text { and } \quad F_{y}=-\frac{G m_{1} m_{2} y}{\left(x^{2}+y^{2}\right)^{3 / 2}} $$ (Hint: First write \(r\) in terms of \(x\) and \(y . )(\text { b) Show that the magnitude}\) of the force on \(m_{2}\) is \(F=G m_{1} m_{2} / r^{2} .\) (c) Does \(m_{1}\) attract or repel \(m_{2} ?\) How do you know?

An object moving in the \(x y\) -plane is acted on by a conservative force described by the potential energy function \(U(x, y)=\) \(\alpha\left(1 / x^{2}+1 / y^{2}\right),\) where \(\alpha\) is a positive constant. Derive an expression for the force expressed in terms of the unit vectors \(\hat{i}\) and \(\hat{y}\) .

The potential energy of two atoms in a diatomic molecule is approximated by \(U(r)=a / r^{12}-b / r^{6},\) where \(r\) is the spacing between atoms and \(a\) and \(b\) are positive constants. (a) Find the force \(F(r)\) one atom as a function of \(r\) . Make two graphs, one of \(U(r)\) versus \(r\) and one of \(F(r)\) versus \(r .\) (b) Find the equilibrium distance between the two atoms. Is this cquilibrium stable? (c) Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b). What minimum energy must be added to the molecule to dissociate it - that is, to separate the two atoms to an infinite distance apart? This is called the dissociation energy of the molecule. (d) For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is \(1.13 \times 10^{-10} \mathrm{m}\) and the dissociation energy is \(1.54 \times 10^{-18} \mathrm{J}\) per molecule. Find the values of the constants \(a\) and \(b\) .

Two blocks with different mass are attached to either end of a light rope that passes over a light, frictionless pulley that is suspended from the ceiling. The masses are released from rest, and the more massive one starts to descend. After this block has descended 1.20 \(\mathrm{m}\) , its speed is 3.00 \(\mathrm{m} / \mathrm{s}\) . If the total mass of the two blocks is \(15.0 \mathrm{kg},\) what is the mass of each block?

Legal Physics. In an auto accident, a car hit a pedestrian and the driver then slammed on the brakes to stop the car. During the subsequent trial, the driver's lawyer claimed that he was obeying the posted 35 \(\mathrm{mi}\) h speed limit, but that the legal speed was too high to allow him to see and react to the pedestrian in time. You have been called in as the state's expert witness. Your investigation of the accident found that the skid marks made while the brakes were applied were 280 \(\mathrm{ft}\) long, and the tread on the tires produced a coefficient of kinetic friction of 0.30 with the road. (a) In your testimony in court, will you say that the driver was obeying the posted speed? You must be able to back up your conclusion with clear reasoning because one of the lawyers will surely cross-examine you. (b) If the driver's speeding ticket were \(\$ 10\) for each mile per hour be was driving above the posted speed limit, would he have to pay a fine? If so, how much would it be?

A \(2.00-\mathrm{kg}\) block is pushed against a spring with negligible mass and force constant \(k=400 \mathrm{N} / \mathrm{m}\) , compressing it 0.220 \(\mathrm{m}\) . When the block is released, it moves along a frictionless, horizontal surface and then up a frictionless incline with slope \(37.0^{\circ}\) (Fig. 7.30\()\) . (a) What is the speed of the block as it slides along the horizontal surface after having left the spring? (b) How far does the block travel up the incline before starting to slide back down?

A block with mass 0.50 \(\mathrm{kg}\) is forced against a horizontal spring of negligible mass, compressing the spring a distance of 0.20 \(\mathrm{m}(\mathrm{Fig} .7 .31) .\) When released, the block moves on a horizontal tabletop for 1.00 \(\mathrm{m}\) before coming to rest. The spring constant \(k\) is 100 \(\mathrm{N} / \mathrm{m}\) . What is the coefficient of kinetic friction \(\mu_{k}\) between the block and the tabletop?

On a horizontal surface, a crate with mass 50.0 \(\mathrm{kg}\) is placed against a spring that stores 360 \(\mathrm{J}\) of energy. The spring is released, and the crate slides 5.60 \(\mathrm{m}\) before coming to rest. What is the speed of the crate when it is 2.00 \(\mathrm{m}\) from its initial position?

Bouncing Ball. A 650 -gram rubber ball is dropped from an initial height of \(250 \mathrm{m},\) and on each bounce it returns to 75\(\%\) of its previous height. (a) What is the initial mechanical energy of the ball, just after it is released from its initial height? (b) How much mechanical energy does the ball lose during its first bounce? What happens to this energy? (c) How much mechanical energy is lost during the second bounce?

Up and Down the Hill. A \(28-\mathrm{kg}\) rock approaches the foot of a hill with a speed of 15 \(\mathrm{m} / \mathrm{s}\) . This hill slopes upward at a constant angle of \(40.0^{\circ}\) above the horizontal. The coefficients of static and kinetic friction between the hill and the rock are 0.75 and 0.20 , respectively. (a) Use energy conservation to find the maximum height above the foot of the hill reached by the rock. (b) Will the rock remain at rest at its highest point, or will it slide back down the hill? (c) If the rock does slide back down, find its speed when it returns to the bottom of the hill.

Bungee Jump. A bungee cord is 30.0 \(\mathrm{m}\) long and, when stretched a distance \(x,\) it exerts a restoring force of magnitude \(k x\) Your father-in- law (mass 95.0 \(\mathrm{kg}\) ) stands on a platform 45.0 \(\mathrm{m}\) above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 \(\mathrm{m}\) before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 380.0 \(\mathrm{N}\) . When you do this, what distance will the bungee cord that you should select have stretched?

Ski Jump Ramp. You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height \(h\) from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of 2.0 \(\mathrm{m} / \mathrm{s}\) as they reach the gate. For safety, the skiers should have a speed of no more than 30.0 \(\mathrm{m} / \mathrm{s}\) when they reach the bottom of the ramp. You determine that for a \(85.0-\mathrm{kg}\) skier with good form, friction and air resistance will do total work of magnitude 4000 \(\mathrm{J}\) on him during his run down the slope. What is the maximum height \(h\) for which the maximum safe speed will not be exceeded?

The Great Sandini is a \(60-\mathrm{kg}\) circus performer who is shot from a cannon (actually a spring gun). You don't find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100 \(\mathrm{N} / \mathrm{m}\) that he will compress with a force of 4400 \(\mathrm{N}\) . The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40 \(\mathrm{N}\) during the 4.0 \(\mathrm{m}\) he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.5 \(\mathrm{m}\) above his initial rest position?

You are designing a delivery ramp for crates containing exercise equipment. The \(1470-\mathrm{N}\) crates will move at 1.8 \(\mathrm{m} / \mathrm{s}\) at the top of a ramp that slopes downward at \(22.0^{\circ} .\) The ramp exerts a \(550-\mathrm{N}\) kinetic friction force on each crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of 8.0 \(\mathrm{m}\) along the ramp. Once stopped, a crate must not rebound back up the ramp. Calculate the force constant of the spring that will be needed in order to meet the design criteria.

A system of two paint buckets connected by a lightweight rope is released from rest with the \(12.0-\mathrm{kg}\) bucket 2.00 \(\mathrm{m}\) above the floor (Fig. 7.36\() .\) Use the principle of conservation of energy to find the speed with which this bucket strikes the floor. You can ignore friction and the mass of the pulley.

A machine part of mass \(m\) is attached to a horizontal ideal spring of force constant \(k\) that is attached to the edge of a friction-free horizontal surface. The part is pushed against the spring compressing it a distance \(x_{0},\) and then released from rest. Find the maximum (a) speed and (b) acceleration of the machine part. (c) Where in the motion do the maxima in parts (a) and (b) occur? (d) What will be the maximum extension of the spring? (e) Describe the subsequent motion of this machine part. Will it ever stop permanently?

A wooden rod of negligible mass and length 80.0 \(\mathrm{cm}\) is pivoted about a horizontal axis through its center. A white rat with mass 0.500 \(\mathrm{kg}\) clings to one end of the stick, and a mouse with mass 0.200 \(\mathrm{kg}\) clings to the other end. The system is released from rest with the rod horizontal. If the animals can manage to hold on, whit are their speeds as the rod swings through a vertical position?

A \(0.100-\mathrm{kg}\) potato is tied to a string with length \(2.50 \mathrm{m},\) and the other end of the string is tied to a rigid support. The potato is held straight out horizontally from the point of support, with the string pulled taut, and is then released. (a) What is the speed of the potato at the lowest point of its motion? (b) What is the tension in the string at this point?

Down the Pole. A fireman of mass \(m\) slides a distance \(d\) down a pole. He starts from rest. He moves as fast at the bottom as if be had stepped off a platform a distance \(h \leq d\) above the ground and descended with negligible air resistance. (a) What average friction force did the fireman exert on the pole? Does your answer make sense in the special cases of \(h=d\) and \(h=0 ?\) (b) Find a numerical value for the average friction force a \(75-\mathrm{kg}\) fireman exerts, for \(d=2.5 \mathrm{m}\) and \(h=1.0 \mathrm{m},(\mathrm{c})\) In terms of \(g, h,\) and \(d\) , what is the speed of the fireman when he is a distance y above the bottom of the pole?

A \(60.0-\mathrm{kg}\) skier starts from rest at the top of a ski slope 65.0 \(\mathrm{m}\) high. (a) If frictional forces do \(-10.5 \mathbf{k J}\) of work on her as she descends, how fast is she going at the bottom of the slope? (b) Now moving horizontally, the skier crosses a patch of soft snow, where \(\mu_{\mathrm{k}}=0.20\) If the patch is 82.0 \(\mathrm{m}\) wide and the average force of air resistance on the skier is 160 \(\mathrm{N}\) , how fast is she going after crossing the patch? (c) The skier hits a snowdrift and Penetrates 2.5 \(\mathrm{m}\) into it before coming to a stop. What is the average force exerted on her by the snowdrift as it stops her?