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Section 6.6 Work (AI6)

Subsection 6.6.1 Activities

Activity 6.6.2.

Consider a bucket with 10 kg of water being pulled against the acceleration of gravity, g=9.8 m/s2, at a constant speed for 20 meters. Using Fact 6.6.1, what is the work needed to pull this bucket up 20 meters in kgm2/s2 (or Nm)?
  1. 10 kgm2/s2
  2. 20 kgm2/s2
  3. 98 kgm2/s2
  4. 200 kgm2/s2
  5. 1960 kgm2/s2

Activity 6.6.3.

Consider the bucket from Activity 6.6.2 with 10 kg of water, being pulled against the acceleration of gravity, g=9.8 m/s2, at a constant speed for 20 meters. Suppose that halfway up at a height of 10m, 5kg of water spilled out, leaving 5kg left. How much total work does it take to get this bucket to a height of 20m?
  1. 980 kgm2/s2 or Nm
  2. 1470 kgm2/s2 or Nm
  3. 1960 kgm2/s2 or Nm

Activity 6.6.4.

Suppose a 10 kg bucket of water is constantly losing water as it’s pulled up, so at a height of h meters, the mass of the bucket is m(h)=2+8e0.2h kg.
Bucket 5 m in the air, to be hoisted by another 5 meters.
Figure 154. Bucket 5 m in the air, to be hoisted by another 5 meters.
(a)
What is the mass of the bucket at height hi=5 m?
(b)
Assuming that the bucket does not lose water, estimate the amount of work needed to lift this bucket up Δh=5 meters.

Activity 6.6.5.

using the same the bucket from Activity 6.6.4, consider the bucket’s mass at heights hi=0,5,10,15 meters.
Bucket raised 5 m at a time, h_i are 0, 5, 10, 15 meters.
Figure 155. Bucket lifted 5 m at a time.
(a)
Fill out the following table, estimating the work it would take to lift the bucket 20 meters.
hiMass m(hi)DistanceEstimated Workh4=15 mm(15)=2+8e0.2152.398 kg5 mh3=10 mm(10)=2+8e0.2103.083 kg5 mh2=5 mm(5)=2+8e0.254.943 kg5 m242.207 Nmh1=0 mm(5)=2+8e0.20=10 kg5 m
(b)
What is the total estimated work to lift this bucket 20 meters?

Activity 6.6.6.

If we estimate the mass and work of the bucket from Activity 6.6.5 at height hi with intervals of length Δh meters, which of the following best represents the Riemann sum of the work it would take to lift this bucket 20 meters?
  1. hi9.8Δh. Nm
  2. (2+8e0.02h)9.8Δm Nm
  3. (2+8e0.02hi)9.8Δh Nm
  4. (2+8e0.02hi)9.8Δm Nm

Activity 6.6.7.

Based on the Riemann sum chosen in Activity 6.6.6, which of the following integrals computes the work it would take to lift this bucket 20 meters?
  1. 020hi9.8dh. Nm
  2. 020(2+8e0.02h)9.8dm Nm
  3. 020(2+8e0.02h)9.8dh Nm
  4. 020(2+8e0.02hi)9.8dh Nm

Activity 6.6.8.

Based on the integral chosen in Activity 6.6.7, compute the work it would take to lift this bucket 20 meters.

Observation 6.6.9. A “how to” for applying integrals to physics.

  1. Estimate the value over a piece of the problem with x value xi over interval of length Δx.
  2. Find a Riemann sum using (1) which estimates the value in question.
  3. Convert the Riemann sum to an integral and solve.

Activity 6.6.10.

Consider a cylindrical tank filled with water, where the base of the cylinder has a radius of 3 meters and a height of 10 meters. Consider a 2 meter thick slice of water sitting 6 meters high in the tank. Using the fact that the mass of this water is 1000π(3)22=18000π kg, estimate how much work is needed to lift this slice 4 more meters to the top of the tank.
2m thick slice of water lifted 4m.
Figure 156. 2m thick slice of water lifted 4m.
  1. 18000π4 Nm
  2. 18000π9.8 Nm
  3. 18000π49.8 Nm
  4. 18000π6 Nm
  5. 18000π69.8 Nm

Activity 6.6.11.

Consider the cylindrical tank filled with water from Activity 6.6.10. We wish to estimate the amount of work required to pump all the water out of the tank. Suppose we slice the water into 5 pieces and estimate the work it would take to lift each piece out of the tank.
2m thick slice of water at 0, 2, 4, 6, 8m.
Figure 157. 2m thick slices of water.
(a)
Fill out the following table, estimating the work it would take to pump all the water out.
hiMassDistanceEstimated Workh5=8 m18000π kgh4=6 m18000π kg4 m705600π Nmh3=4 m18000π kgh2=2 m18000π kgh1=0 m18000π kg10 m
(b)
What is the total estimated work to pump out all the water?

Activity 6.6.12.

Recall Activity 6.6.11. If we estimate the work needed to lift slices of thickness Δh m at heights hi m, which of the following Riemann sums best estimates the total work needed to pump all the water from the tank?
  1. 1000π329.8(10h)Δh Nm
  2. 1000π329.8(10hi)Δh Nm
  3. 1000π(hi)29.8(10h)Δh Nm
  4. 1000π(hi)29.8(10hi)Δh Nm

Activity 6.6.13.

Based on the Riemann sum chosen in Activity 6.6.12, which of the following integrals computes the work it would take to pump all the water from the tank?
  1. 0109000π9.8(10h)dh Nm
  2. 0101000π9.8h2(10h)dh Nm

Activity 6.6.14.

Based on the integral chosen in Activity 6.6.13, compute the work it would take to pump all the water out of the tank.

Activity 6.6.15.

Consider a cylindrical truncated-cone tank where the radius on the bottom of the tank is 10 m, the radius at the top of the tank is 100 m, and the height of the tank is 100m.
described in detail following the image
A slice at height hi of width Δh, with radius ri.
Figure 158. A slice at height hi of width Δh.
(a)
What is the radius ri in meters of the cross section made at height hi meters?
(b)
What is the volume of a cylinder with radius ri meters with width Δh meters?
(c)
Using the fact that water has density 1000 kg/m3, what is the mass of the volume of water you found in (b)?
(d)
How far must this cylinder of water be lifted to be out of the tank?

Activity 6.6.16.

Recall the computations done in Activity 6.6.15.
(a)
Find a Riemann sum which estimates the total work needed to pump all the water out of this tank, using slices at heights hi m, of width Δh m.
(b)
Use (a) to find an integral expression which computes the amount of work needed to pump all the water out of this tank.
(c)
Evaluate the integral found in (b).

Subsection 6.6.2 Videos

Figure 159. Video: Set up integrals to solve problems involving work, force, and/or pressure

Subsection 6.6.3 Exercises