Chapter 8

(I) A spring has a spring constant \(k\) of \(82.0 \mathrm{~N} / \mathrm{m} .\) How much must this spring be compressed to store \(35.0 \mathrm{~J}\) of potential energy?

(1) A spring has a spring constant \(k\) of 82.0 \(\mathrm{N} / \mathrm{m} .\) How much must this spring be compressed to store 35.0 \(\mathrm{J}\) of potential energy?

(I) A 6.0-kg monkey swings from one branch to another \(1.3 \mathrm{~m}\) higher. What is the change in gravitational potential energy?

(II) A spring with \(k=63 \mathrm{~N} / \mathrm{m}\) hangs vertically next to a ruler. The end of the spring is next to the \(15-\mathrm{cm}\) mark on the ruler. If a \(2.5-\mathrm{kg}\) mass is now attached to the end of the spring, where will the end of the spring line up with the ruler marks?

(II) A \(56.5-\mathrm{kg}\) hiker starts at an elevation of \(1270 \mathrm{~m}\) and climbs to the top of a 2660-m peak. (a) What is the hiker's change in potential energy? (b) What is the minimum work required of the hiker? \((c)\) Can the actual work done be greater than this? Explain.

(II) A 56.5 -kg hiker starts at an elevation of 1270 \(\mathrm{m}\) and climbs to the top of a \(2660-\mathrm{m}\) peak. (a) What is the hiker's change in potential energy? (b) What is the minimum work required of the hiker? (c) Can the actual work done be greater than this? Explain.

(II) A 1.60-m tall person lifts a 1.95 -kg book off the ground so it is \(2.20 \mathrm{~m}\) above the ground. What is the potential energy of the book relative to \((a)\) the ground, and \((b)\) the top of the person's head? \((c)\) How is the work done by the person related to the answers in parts \((a)\) and \((b) ?\)

(II) A \(1.60-\mathrm{m}\) tall person lifts a 1.95 -kg book off the ground so it is 2.20 \(\mathrm{m}\) above the ground. What is the potential energy of the book relative to \((a)\) the ground, and \((b)\) the top of the person's head? (c) How is the work done by the person related to the answers in parts \((a)\) and \((b) ?\)

(II) A 1200 -kg car rolling on a horizontal surface has speed \(v=75 \mathrm{~km} / \mathrm{h}\) when it strikes a horizontal coiled spring and is brought to rest in a distance of \(2.2 \mathrm{~m}\). What is the spring stiffness constant of the spring?

(II) A \(1200-\mathrm{kg}\) car rolling on a horizontal surface has speed \(v=75 \mathrm{km} / \mathrm{h}\) when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 \(\mathrm{m} .\) What is the spring stiffness constant of the spring?

(II) A particular spring obeys the force law \(\overrightarrow{\mathbf{F}}=\) \(\left(-k x+a x^{3}+b x^{4}\right) \hat{\mathbf{i}} . \quad(a)\) Is this force conservative? Explain why or why not. \((b)\) If it is conservative, determine the form of the potential energy function.

(1I) A particular spring obeys the force law \(\vec{\mathbf{F}}=\) \(\left(-k x+a x^{3}+b x^{4}\right) \hat{\mathbf{i}}\) (a) Is this force conservative? Explain why or why not. (b) If it is conservative, determine the form of the potential energy function.

(II) A particle is constrained to move in one dimension along the \(x\) axis and is acted upon by a force given by \(\overrightarrow{\mathbf{F}}(x)=-\frac{k}{x^{3}} \hat{\mathbf{i}}\) where \(k\) is a constant with units appropriate to the SI system. Find the potential energy function \(U(x)\), if \(U\) is arbitrarily defined to be zero at \(x=2.0 \mathrm{~m},\) so that \(U(2.0 \mathrm{~m})=0 .\)

(II) A particle constrained to move in one dimension is subject to a force \(F(x)\) that varies with position \(x\) as $$\overrightarrow{\mathbf{F}}(x)=A \sin (k x) \hat{\mathbf{i}}$$ where \(A\) and \(k\) are constants. What is the potential energy function \(U(x),\) if we take \(U=0\) at the point \(x=0 ?\)

(I) A novice skier, starting from rest, slides down a frictionless \(13.0^{\circ}\) incline whose vertical height is \(125 \mathrm{~m} .\) How fast is she going when she reaches the bottom?

(I) Jane, looking for Tarzan, is running at top speed \((5.0 \mathrm{~m} / \mathrm{s})\) and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does the length of the vine affect your answer?

(II) In the high jump, the kinetic energy of an athlete is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must the athlete leave the ground in order to lift his center of mass \(2.10 \mathrm{~m}\) and cross the bar with a speed of \(0.70 \mathrm{~m} / \mathrm{s} ?\)

(II) A sled is initially given a shove up a frictionless \(23.0^{\circ}\) incline. It reaches a maximum vertical height \(1.12 \mathrm{~m}\) higher than where it started. What was its initial speed?

(1I) A 55 -kg bungee jumper leaps from a bridge. She is tied to a bungee cord that is 12 \(\mathrm{m}\) long when unstretched, and falls a total of 31 \(\mathrm{m}\) . (a) Calculate the spring constant \(k\) of the bungee cord assuming Hooke's law applies. (b) Calcu- late the maximum acceleration she experiences.

(II) A 72 -kg trampoline artist jumps vertically upward from the top of a platform with a \(\begin{array}{llll}\text { speed of } & 4.5 \mathrm{~m} / \mathrm{s} . & (a) & \text { How }\end{array}\) fast is he going as he lands on the trampoline, \(2.0 \mathrm{~m}\) below (Fig. \(8-31\) )? (b) If the trampoline behaves like a spring of spring constant \(5.8 \times 10^{4} \mathrm{~N} / \mathrm{m}\), how far does he depress it?

(II) \(\mathrm{A} 72\) -kg trampoline artist jumps vertically upward from the top of a platform with a speed of 4.5 \(\mathrm{m} / \mathrm{s}\) . (a) How fast is he going as he lands on the trampoline, 2.0 \(\mathrm{m}\) below (Fig. 31\() ?\) (b) If the trampoline behaves like a spring of spring constant \(5.8 \times 10^{4} \mathrm{N} / \mathrm{m},\) how far does he depress it?

(II) The total energy \(E\) of an object of mass \(m\) that moves in one dimension under the influence of only conservative forces can be written as $$E=\frac{1}{2} m v^{2}+U$$ Use conservation of energy, \(d E / d t=0,\) to predict Newton's second law.

(II) A \(0.40-\mathrm{kg}\) ball is thrown with a speed of \(8.5 \mathrm{~m} / \mathrm{s}\) at an upward angle of \(36^{\circ} .(a)\) What is its speed at its highest point, and \((b)\) how high does it go? (Use conservation of energy.)

(II) \(\mathrm{A}\) 0.40-kg ball is thrown with a speed of 8.5 \(\mathrm{m} / \mathrm{s}\) at an upward angle of \(36^{\circ} .(a)\) What is its speed at its highest point, and \((b)\) how high does it go? (Use conservation of energy.)

(II) A vertical spring (ignore its mass), whose spring constant is \(875 \mathrm{~N} / \mathrm{m},\) is attached to a table and is compressed down by \(0.160 \mathrm{~m} .\) ( \(a\) ) What upward speed can it give to a \(0.380-\mathrm{kg}\) ball when released? \((b)\) How high above its original position (spring compressed) will the ball fly?

(II) A block of mass \(m\) is attached to the end of a spring (spring stiffness constant \(k\) ), Fig. 8-35. The mass is given an initial displacement \(x_{0}\) from equilibrium, and an initial speed \(v_{0}\). Ignoring friction and the mass of the spring, use energy methods to find \((a)\) its maximum speed, and \((b)\) its maximum stretch from equilibrium, in terms of the given quantities.

(II) A cyclist intends to cycle up a \(9.50^{\circ}\) hill whose vertical height is \(125 \mathrm{~m}\). The pedals turn in a circle of diameter \(36.0 \mathrm{~cm} .\) Assuming the mass of bicycle plus person is \(75.0 \mathrm{~kg},\) (a) calculate how much work must be done against gravity. (b) If each complete revolution of the pedals moves the bike \(5.10 \mathrm{~m}\) along its path, calculate the average force that must be exerted on the pedals tangent to their circular path. Neglect work done by friction and other losses.

(1I) A cyclist intends to cycle up a \(9.50^{\circ}\) hill whose vertical height is 125 \(\mathrm{m}\) . The pedals turn in a circle of diameter 36.0 \(\mathrm{cm} .\) Assuming the mass of bicycle plus person is 75.0 \(\mathrm{kg}\) , (a) calculate how much work must be done against gravity. (b) If each complete revolution of the pedals moves the bike 5.10 \(\mathrm{m}\) along its path, calculate the average force that must be exerted on the pedals tangent to their circular path. Neglect work done by friction and other losses.

(II) A pendulum \(2.00 \mathrm{~m}\) long is released (from rest) at an angle \(\theta_{0}=30.0^{\circ}\) (Fig. \(8-14\) ). Determine the speed of the \(70.0-\mathrm{g}\) bob: \((a)\) at the lowest point \((\theta=0) ;(b)\) at \(\theta=15.0^{\circ}\) (c) at \(\theta=-15.0^{\circ}\) (i.e., on the opposite side). ( \(d\) ) Determine the tension in the cord at each of these three points. \((e)\) If the bob is given an initial speed \(v_{0}=1.20 \mathrm{~m} / \mathrm{s}\) when released at \(\theta=30.0^{\circ},\) recalculate the speeds for parts \((a),(b),\) and \((c)\)

(II) What should be the spring constant \(k\) of a spring designed to bring a \(1200-\mathrm{kg}\) car to rest from a speed of \(95 \mathrm{~km} / \mathrm{h}\) so that the occupants undergo a maximum acceleration of \(5.0 g ?\)

(I) Two railroad cars, each of mass \(56,000 \mathrm{~kg}\), are traveling \(95 \mathrm{~km} / \mathrm{h}\) toward each other. They collide head-on and come to rest. How much thermal energy is produced in this collision?

(I) Tiwo railroad cars, each of mass \(56,000 \mathrm{kg}\) , are traveling 95 \(\mathrm{km} / \mathrm{h}\) toward each other. They collide head-on and come to rest. How much thermal energy is produced in this collision?

(I) A \(16.0-\mathrm{kg}\) child descends a slide \(2.20 \mathrm{~m}\) high and reaches the bottom with a speed of \(1.25 \mathrm{~m} / \mathrm{s}\). How much thermal energy due to friction was generated in this process?

(1) A 16.0 -kg child descends a slide 2.20 \(\mathrm{m}\) high and reaches the bottom with a speed of 1.25 \(\mathrm{m} / \mathrm{s} .\) How much thermal energy due to friction was generated in this process?

(II) A ski starts from rest and slides down a \(28^{\circ}\) incline \(85 \mathrm{~m}\) long. \((a)\) If the coefficient of friction is \(0.090,\) what is the ski's speed at the base of the incline? \((b)\) If the snow is level at the foot of the incline and has the same coefficient of friction, how far will the ski travel along the level? Use energy methods.

(II) A \(145-\mathrm{g}\) baseball is dropped from a tree \(14.0 \mathrm{~m}\) above the ground. ( \(a\) ) With what speed would it hit the ground if air resistance could be ignored? \((b)\) If it actually hits the ground with a speed of \(8.00 \mathrm{~m} / \mathrm{s}\), what is the average force of air resistance exerted on it?

(II) A 145 -g baseball is dropped from a tree 14.0 \(\mathrm{m}\) above the ground. \((a)\) With what speed would it hit the ground if air resistance could be ignored? (b) If it actually hits the ground with a speed of \(8.00 \mathrm{m} / \mathrm{s},\) what is the average force of air resistance exerted on it?

(II) A 96-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of \(350 \mathrm{~N}\). For the first \(15 \mathrm{~m}\) the floor is frictionless, and for the next \(15 \mathrm{~m}\) the coefficient of friction is \(0.25 .\) What is the final speed of the crate?

(II) A \(96-\mathrm{kg}\) crate, starting from rest, is pulled across a floor with a constant horizontal force of 350 \(\mathrm{N}\) . For the first 15 \(\mathrm{m}\) the floor is frictionless, and for the next 15 \(\mathrm{m}\) the coefficient of friction is \(0.25 .\) What is the final speed of the crate?

(II) A skier traveling \(9.0 \mathrm{~m} / \mathrm{s}\) reaches the foot of a steady upward \(19^{\circ}\) incline and glides \(12 \mathrm{~m}\) up along this slope before coming to rest. What was the average coefficient of friction?

(II) A 0.620 -kg wood block is firmly attached to a very light horizontal spring \((k=180 \mathrm{~N} / \mathrm{m})\) as shown in Fig. \(8-35 .\) This block-spring system, when compressed \(5.0 \mathrm{~cm}\) and released, stretches out \(2.3 \mathrm{~cm}\) beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table?

(II) A \(180-\mathrm{g}\) wood block is firmly attached to a very light horizontal spring, Fig. \(8-35 .\) The block can slide along a table where the coefficient of friction is \(0.30 .\) A force of \(25 \mathrm{~N}\) compresses the spring \(18 \mathrm{~cm}\). If the spring is released from this position, how far beyond its equilibrium position will it stretch on its first cycle?

(II) A 180 -g wood block is firmly attached to a very light horizontal spring, Fig. \(35 .\) The block can slide along a table where the coefficient of friction is \(0.30 .\) A force of 25 \(\mathrm{N}\) compresses the spring 18 \(\mathrm{cm} .\) If the spring is released from this position, how far beyond its equilibrium position will it stretch on its first cycle?

(II) You drop a ball from a height of \(2.0 \mathrm{~m}\), and it bounces back to a height of \(1.5 \mathrm{~m}\). ( \(a\) ) What fraction of its initial energy is lost during the bounce? (b) What is the ball's speed just before and just after the bounce? ( \(c\) ) Where did the energy go?

You drop a ball from a height of \(2.0 \mathrm{m},\) and it bounces back to a height of 1.5 \(\mathrm{m}\) (a) What fraction of its initial energy is lost during the bounce? (b) What is the ball's speed just before and just after the bounce? (c) Where did the energy go?

(II) A 56-kg skier starts from rest at the top of a 1200 -mlong trail which drops a total of \(230 \mathrm{~m}\) from top to bottom. At the bottom, the skier is moving \(11.0 \mathrm{~m} / \mathrm{s}\). How much energy was dissipated by friction?

(1I) A 56 -kg skier starts from rest at the top of a 1200 -m- long trail which drops a total of 230 m from top to bottom. At the bottom, the skier is moving 11.0 \(\mathrm{m} / \mathrm{s} .\) How much. energy was dissignated by friction?

(III) A spring \((k=75 \mathrm{N} / \mathrm{m})\) has an equilibrium length of 1.00 \(\mathrm{m} .\) The spring is compressed to a length of 0.50 \(\mathrm{m}\) and a mass of 2.0 \(\mathrm{kg}\) is placed at its free end on a frictionless slope which makes an angle of \(41^{\circ}\) with respect to the horizontal (Fig. \(38 ) .\) The spring is then released. (a) If the mass is not attached to the spring, how far up the slope will the mass move before coming to rest? (b) If the mass is attached to the spring, how far up the slope will the mass move before coming to rest? (c) Now the incline has a coefficient of kinetic friction \(\mu_{k}\) . If the block, attached to the spring, is observed to stop just as it reaches the spring's equilibrium position, what is the coefficient of friction \(\mu_{k} ?\)

(III) A \(2.0-\mathrm{kg}\) block slides along a horizontal surface with a coefficient of kinetic friction \(\mu_{\mathrm{k}}=0.30 .\) The block has a speed \(v=1.3 \mathrm{~m} / \mathrm{s}\) when it strikes a massless spring head- on (as in Fig. \(8-18\) ). \((a)\) If the spring has force constant \(k=120 \mathrm{~N} / \mathrm{m},\) how far is the spring compressed? (b) What minimum value of the coefficient of static friction, \(\mu_{S}\), will assure that the spring remains compressed at the maximum compressed position? (c) If \(\mu_{\mathrm{S}}\) is less than this, what is the speed of the block when it detaches from the decompressing spring? [Hint: Detachment occurs when the spring reaches its natural length \((x=0) ;\) explain why.

(III) \(\mathrm{A} 2.0\) -kg block slides along a horizontal surface with a coefficient of kinetic friction \(\mu_{\mathrm{k}}=0.30 .\) The block has a speed \(v=1.3 \mathrm{m} / \mathrm{s}\) when it strikes a massless spring head- on. (a) If the spring has force constant \(k=120 \mathrm{N} / \mathrm{m},\) how far is the spring compressed? (b) What minimum value of the coefficient of static friction, \(\mu_{\mathrm{S}},\) will assure that the spring remains compressed at the maximum compressed position? (c) If \(\mu_{\mathrm{s}}\) is less than this, what is the speed of the block when it detaches from the decompressing spring? [Hint: Detach- ment occurs when the spring reaches its natural length \((x=0) :\) explain why 1