Stewart 8th §6.4: Work

39 questions

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Stewart 8th §6.4: Work 0/39
1 Work · Level 2
A 360-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in
(a) 10 seconds
(b) 5 seconds

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2 Work · Level 2
How much work is done when a hoist lifts a 200-kg rock to a height of 3 m?
3 Work · Level 2
A variable force of \(5x^{-2}\) pounds moves an object along a straight line when it is \(x\) feet from the origin. Calculate the work done in moving the object from \(x = 1\) ft to \(x = 10\) ft.
4 Work · Level 3
When a particle is located a distance \(x\) meters from the origin, a force of \(\cos\left(\dfrac{\pi x}{3}\right)\) newtons acts on it. How much work is done in moving the particle from \(x = 1\) to \(x = 2\)? Interpret your answer by considering the work done from \(x = 1\) to \(x = 1.5\) and from \(x = 1.5\) to \(x = 2\).
5 Work · Level 3
Shown is the graph of a force function (in newtons) that increases to its maximum value and then remains constant. How much work is done by the force in moving an object a distance of 8 m? [Figure: force increases linearly from 0 to 30 N over first 4 m, then constant at 30 N]
6 Work - Midpoint Rule · Level 3
The table shows values of a force function \(f(x)\), where \(x\) is measured in meters and \(f(x)\) in newtons. Use the Midpoint Rule to estimate the work done by the force in moving an object from \(x = 4\) to \(x = 20\).
\(x\) 4 6 8 10 12 14 16 18 20
\(f(x)\) 5 5.8 7.0 8.8 9.6 8.2 6.7 5.2 4.1
7 Work - Hooke's Law · Level 2
A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?
8 Work - Hooke's Law · Level 3
A spring has a natural length of 40 cm. If a 60-N force is required to keep the spring compressed 10 cm, how much work is done during this compression? How much work is required to compress the spring to a length of 25 cm?
9 Work - Hooke's Law · Level 3
Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm.
(a) How much work is needed to stretch the spring from 35 cm to 40 cm?
(b) How far beyond its natural length will a force of 30 N keep the spring stretched?

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10 Work - Hooke's Law · Level 3
If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?
11 Work - Hooke's Law · Level 3
A spring has natural length 20 cm. Compare the work \(W_1\) done in stretching the spring from 20 cm to 30 cm with the work \(W_2\) done in stretching it from 30 cm to 40 cm. How are \(W_2\) and \(W_1\) related?
12 Work - Hooke's Law · Level 3
If 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring?
13 Work - Cable/Rope · Level 3
A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high.
(a) How much work is done in pulling the rope to the top of the building?
(b) How much work is done in pulling half the rope to the top of the building?

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14 Work - Cable/Rope · Level 3
A thick cable, 60 ft long and weighing 180 lb, hangs from a winch on a crane. Compute in two different ways the work done if the winch winds up 25 ft of the cable.
(a) Follow the method of Example 4.
(b) Write a function for the weight of the remaining cable after \(x\) feet has been wound up by the winch. Estimate the amount of work done when the winch pulls up \(\Delta x\) ft of cable.

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15 Work - Cable/Rope · Level 3
A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.
16 Work - Cable/Rope · Level 3
A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m?
17 Work - Cable/Rope · Level 3
A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?
18 Work - Cable/Rope · Level 3
A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.
19 Work - Cable/Rope · Level 3
A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that it's level with the upper end.
20 Work - Pumping · Level 3
A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5 lb/ft\(^3\).)
21 Work - Pumping · Level 3
An aquarium 2 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is 1000 kg/m\(^3\).)
22 Work - Pumping · Level 4
A spherical water tank, 24 ft in diameter, sits atop a 60 ft tower. The tank is filled by a hose attached to the bottom of the sphere. If a 1.5 horsepower pump is used to deliver water up to the tank, how long will it take to fill the tank? (One horsepower \(= 550\) ft-lb of work per second.)
23 Work - Pumping · Level 3
A tank is full of water. Find the work required to pump the water out of the spout. [Figure: rectangular tank 3 m wide, 5 m long, 3 m deep with 2 m spout on top]
24 Work - Pumping · Level 3
A tank is full of water. Find the work required to pump the water out of the spout. [Figure: trapezoidal cross-section tank 8 m long with 1 m spout, 3 m at top and wider at bottom]
25 Work - Pumping · Level 3
A tank is full of water. Find the work required to pump the water out of the spout. [Figure: frustum of a cone, top 6 ft, bottom 3 ft, height 8 ft]
26 Work - Pumping · Level 3
A tank is full of water. Find the work required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft\(^3\). [Figure: inverted frustum of a cone, top 12 ft, bottom 6 ft, height 10 ft]
27 Work - Pumping · Level 4
Suppose that for the tank in Exercise 23 the pump breaks down after \(4.7 \times 10^5\) J of work has been done. What is the depth of the water remaining in the tank?
28 Work - Pumping · Level 3
Solve Exercise 24 if the tank is half full of oil that has a density of 900 kg/m\(^3\).
29 Work · Level 4
When gas expands in a cylinder with radius \(r\), the pressure at any given time is a function of the volume: \(P = P(V)\). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: \(F = \pi r^2 P\). Show that the work done by the gas when the volume expands from volume \(V_1\) to volume \(V_2\) is \(W = \displaystyle\int_{V_1}^{V_2} P d V\).
30 Work · Level 4
In a steam engine the pressure \(P\) and volume \(V\) of steam satisfy the equation \(P V^{1.4} = k\), where \(k\) is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Use Exercise 29 to calculate the work done by the engine during a cycle when the steam starts at a pressure of 160 lb/in\(^2\) and a volume of 100 in\(^3\) and expands to a volume of 800 in\(^3\).
31 Work - Kinetic Energy · Level 4
The kinetic energy KE of an object of mass \(m\) moving with velocity \(v\) is defined as \(\text{KE} = \dfrac{1}{2} m v^2\). If a force \(f(x)\) acts on the object, moving it along the \(x\)-axis from \(x_1\) to \(x_2\), the Work-Energy Theorem states that the net work done is equal to the change in kinetic energy: \(\dfrac{1}{2} m v_2^2 - \dfrac{1}{2} m v_1^2\).
(a) Prove the Work-Energy Theorem.
(b) How much work (in ft-lb) is required to hurl a 12-lb bowling ball at a speed of 20 mi/h?

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32 Work · Level 4
Suppose that when launching an 800-kg roller coaster car an electromagnetic propulsion system exerts a force of \(5.7x^2 + 1.5x\) newtons on the car at a distance \(x\) meters along the track. Use Exercise 31(a) to find the speed of the car when it has traveled 60 meters.
33 Work - Gravity · Level 4
(a) Newton's Law of Gravitation states that two bodies with masses \(m_1\) and \(m_2\) attract each other with a force \(F = G \dfrac{m_1 m_2}{r^2}\) where \(r\) is the distance between the bodies and \(G\) is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from \(r = a\) to \(r = b\).
(b) Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earth's mass is \(5.98 \times 10^{24}\) kg and is concentrated at its center. Take the radius of the earth to be \(6.37 \times 10^6\) m and \(G = 6.67 \times 10^{-11}\) N\(\cdot\)m\(^2\)/kg\(^2\).

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34 Work · Level 5
The Great Pyramid of King Khufu was built of limestone over a 20-year time period from 2580 BC to 2560 BC. Its base is a square with side length 756 ft and its height when built was 481 ft. (It was the tallest man-made structure in the world for more than 3800 years.) The density of the limestone is about 150 lb/ft\(^3\).
(a) Estimate the total work done in building the pyramid.
(b) If each laborer worked 10 hours a day for 340 days a year, and did 200 ft-lb/h of work in lifting the limestone blocks into place, about how many laborers were needed to construct the pyramid?

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35 Work · Level 1
(a) How much work is done in lifting a 1.2-kg book off the floor to put it on a desk that is 0.7 m high? Use the fact that the acceleration due to gravity is \(g = 9.8\) m/s\(^2\).
(b) How much work is done in lifting a 20-lb weight 6 ft off the ground?

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36 Work · Level 2
When a particle is located a distance \(x\) feet from the origin, a force of \(x^2 + 2x\) pounds acts on it. How much work is done in moving it from \(x = 1\) to \(x = 3\)?
37 Work - Hooke's Law · Level 3
A force of 40 N is required to hold a spring that has been stretched from its natural length of 10 cm to a length of 15 cm. How much work is done in stretching the spring from 15 cm to 18 cm?
38 Work - Cable/Rope · Level 3
A 200-lb cable is 100 ft long and hangs vertically from the top of a tall building. How much work is required to lift the cable to the top of the building?
39 Work - Pumping · Level 4
A tank has the shape of an inverted circular cone with height 10 m and base radius 4 m. It is filled with water to a height of 8 m. Find the work required to empty the tank by pumping all of the water to the top of the tank. (The density of water is 1000 kg/m\(^3\).)

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