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Chapter 11: Problem 63

What force must the surface of a basketball withstand if it is inflated to a pressure of \(8.5\) psi? Assume the ball has a diameter of \(23 \mathrm{~cm}\).

Short Answer

Expert verified
The force the surface of the basketball must withstand is approximately 9730.51 N.

Step by step solution

01

Convert pressure from psi to pascals

1 psi is approximately 6894.76 pascals. Therefore, convert the given pressure of 8.5 psi to pascals by multiplying it by 6894.76: \(Pressure (Pa) = 8.5 \, psi * 6894.76 \, Pa/psi = 58605.46 \, Pa.\)
02

Calculate the radius of the basketball

The radius of the ball is half the diameter, so divide the diameter by 2: \(Radius = Diameter / 2 = 23 \, cm / 2 = 11.5 \, cm.\) However, the pressure is in pascals, and pascals are a unit of pressure in the International System of Units (SI), so the radius must be in meters (m). To convert cm to m, divide by 100: \(Radius (m) = 11.5 \, cm / 100 = 0.115 \, m.\)
03

Calculate the surface area of the basketball

The surface area formula for a sphere is \(A = 4 * \pi * r^2.\) Substituting the value of radius obtained in step 2, \(A = 4 * \pi * (0.115 \, m)^2 = 0.166 \, m^2.\)
04

Calculate the force exerted by the air inside the basketball

The force can be calculated using the formula: Force = Pressure * Area. Substituting the values from step 1 and 3: \(Force = 58605.46 \, Pa * 0.166 \, m^2 = 9730.51 \, N.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pressure Conversion
When working with different systems of measurement, converting units accurately is crucial. Pressure can be applied in various units such as psi (pounds per square inch) and pascals (Pa). Often, problems will provide pressure measurements in psi since it's commonly used in the US.

To convert from psi to pascals, use the conversion factor where 1 psi is approximately equal to 6894.76 Pa. Thus, to convert 8.5 psi to pascals, multiply:
  • Pressure in pascals = 8.5 * 6894.76 = 58605.46 Pa.
This conversion ensures consistency with the International System of Units (SI), which is essential for accurate calculations in physics problems.
Surface Area Calculation
To find the force exerted on a surface, calculating the surface area is necessary. In this scenario, since we are dealing with a sphere (a basketball), the formula used is for the surface area of a sphere:
  • Surface Area, A = \(4 \pi r^2\), where \(r\) is the radius.
First, find the basketball's radius, which is half of its diameter (23 cm). Then convert this value into meters by dividing by 100, giving 0.115 meters.

Substituting in the formula:
  • A = \(4 \cdot \pi \cdot (0.115 \, m)^2\) results in approximately 0.166 square meters.
The surface area calculation is fundamental to understand how distributed forces affect the basketball's surface.
Force Calculation
Once the pressure and surface area are known, you can calculate the force exerted on the object. Force calculation is done by multiplying the pressure by the surface area. This is based on the formula:
  • \(\text{Force} = \text{Pressure} \times \text{Area}\).
Using the previously calculated values:
  • Pressure: 58605.46 Pa
  • Surface Area: 0.166 square meters
We find that:
  • \(\text{Force} = 58605.46 \, Pa \times 0.166 \, m^2 = 9730.51 \, N\).
This result tells us that the basketball must withstand a force of approximately 9730.51 Newtons, highlighting the importance of each calculation step in solving physics problems effectively.

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Most popular questions from this chapter

A crown that is supposed to be made of solid gold is under suspicion. When the crown is weighed in air it has a weight of \(5.15 \mathrm{~N}\). When it is suspended from a digital balance and lowered into water, its apparent weight is measured to be \(4.88 \mathrm{~N}\). Given that the specific gravity of gold is \(19.3\), comment on the authenticity of the crown. SSM

Medical The aorta is approximately \(25 \mathrm{~mm}\) in diameter. The mean pressure there is about \(100 \mathrm{mmHg}\) and the blood flows through the aorta at approximately \(60 \mathrm{~cm} / \mathrm{s}\). Suppose that at a certain point a portion of the aorta is blocked so that the cross-sectional area is reduced to \(3 / 4\) of its original area. The density of blood is \(1060 \mathrm{~kg} / \mathrm{m}^{3}\). (a) How fast is the blood moving just as it enters the blocked portion of the aorta? (b) What is the gauge pressure (in \(\mathrm{mmHg}\) ) of the blood just as it has entered the blocked portion of the aorta?

Astronomy When a star reaches the end of its life it is possible for a supernova to occur. This may result in the formation of a very small, but very dense, neutron star. The density of such a stellar remnant is about the same as a neutron. Determine the radius of a neutron star that has the mass of our Sun and the same density as a neutron. A neutron has a mass of \(1.7 \times 10^{-27} \mathrm{~kg}\) and an approximate radius of \(1.2 \times 10^{-15} \mathrm{~m}\). Assume that both the neutron star and the neutron itself are spherical in shape. SSM

Medical Blood flows through an artery that is partially blocked. As the blood moves from the wider region into the narrow region, the blood speed A. increases. B. decreases. C. stays the same. D. drops to zero. E. alternately increases and then decreases.

Medical Suppose that your pressure gauge for determining the blood pressure of a patient measured absolute pressure instead of gauge pressure. How would you write the normal value of systolic blood pressure, \(120 \mathrm{mmHg}\), in such a case? SSM
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