# Mechanics: Magnitude of a buoyant force on the sphere

Problem 93

A 7.49 kg sphere of radius 5.25 cm is at a depth of 2.20 km in seawater that has an average density of 1025 kg/m3. What are the (a) gauge pressure, (b) total pressure, and (c) corresponding total force compressing the sphere's surface? What are (d) the magnitude of the buoyant force on the sphere and (e) the magnitude of the sphere's acceleration if it is free to move? Take atmospheric pressure to be 1.01 x 105 Pa.

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#### Solution Preview

A 7.49 kg sphere of radius 5.25 cm is at a depth of 2.20 km in seawater that has an average density of 1025 kg/m3. What are the (a) gauge pressure, (b) total pressure, and (c) corresponding total force compressing the sphere's surface? What are (d) the magnitude of the buoyant force on the sphere and (e) the magnitude of the sphere's acceleration if it is free to move? Take atmospheric pressure to be 1.01 x 105 Pa.

Solution:

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#### Solution Summary

The magnitude of a buoyant force on the sphere is calculated using mechanics.

Rotational mechanics, hydrostatics and hydrodynamics.

1 The arm in Figure P8.7 weighs 36.0 N. The force of gravity acting on the arm acts through point A. Determine the magnitudes of the tension force t in the deltoid muscle and the force s of the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown.

2. One end of a uniform 6.0 m long rod of weight w is supported by a cable. The other end rests against the wall, where it is held by friction (see Fig. P8.28). The coefficient of static friction between the wall and the rod is µs = 0.50. Determine the minimum distance, x, from point A at which an additional weight w (the same as the weight of the rod) can be hung without causing the rod to slip at point A.

3. The puck in Figure P8.51 has a mass of 0.220 kg. Its original distance from the center of rotation is 40.0 cm, and it moves with a speed of 65.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. Determine the work done on the puck. (Hint: Consider the change of kinetic energy of the puck.)

4. A 230 N sphere 0.20 m in radius rolls, without slipping 6.0 m down a ramp that is inclined at 28° with the horizontal. What is the angular speed of the sphere at the bottom of the hill if it starts from rest?

rad/s

5. Use conservation of energy to determine the angular speed of the spool shown in Figure P8.36 after the 3.00 kg bucket has fallen 4.45 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

6. The total cross-sectional area of the load-bearing calcified portion of the two forearm bones (radius and ulna) is approximately 2.3 cm2. During a car crash, the forearm is slammed against the dashboard. The arm comes to rest from an initial speed of 80 km/h in 4.5 ms. If the arm has an effective mass of 3.0 kg, what is the compressional stress that the arm withstands during the crash?

7. The density of ice is 920 kg/m3, and that of seawater is 1030 kg/m3. What fraction of the total volume of an iceberg is exposed?

8. A 9.0 kg block of metal is suspended from a scale and immersed in water as in Figure P9.30. The dimensions of the block are 12.0 cm 9.0 cm 9.0 cm. The 12.0 cm dimension is vertical, and the top of the block is 5.00 cm below the surface of the water.

(a) What are the forces exerted by the water on the top and bottom of the block? (Do not ignore the effect of the air above the water. Take P0 = 1.0130 105 N/m2.)

(b) What is the reading of the spring scale?

(c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block.

9. A hypodermic syringe contains a medicine with the density of water (Fig. P9.43). The barrel of the syringe has a cross-sectional area of 2.70 10-5 m2. In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force, , of magnitude 2.20 N is exerted on the plunger, making medicine squirt from the needle. Determine the medicine's flow speed through the needle. Assume that the pressure in the needle remains equal to 1.00 atm and that the syringe is horizontal.

10. Water is pumped through a pipe of diameter 15.0 cm from the Colorado River up to a village on the rim of the canyon. The river is at 564 m elevation and the village is at 2149 m.

(a) At what minimum pressure must the water be pumped to arrive at the village?

(b) If 5000 m3 are pumped per day, what is the speed of the water in the pipe?

(c) What additional pressure is necessary to deliver this flow? [Note: You may assume that the free-fall acceleration and the density of air are constant over the given range of elevations.]