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Chapter 8: Problem 12

Experimental tests have shown that bone will rupture if it is subjected to a force density of 1.03 \(\times\) 10\(^8\) N/m\(^2\). Suppose a 70.0-kg person carelessly rollerskates into an overhead metal beam that hits his forehead and completely stops his forward motion. If the area of contact with the person's forehead is 1.5 cm\(^2\), what is the greatest speed with which he can hit the wall without breaking any bone if his head is in contact with the beam for 10.0 ms?

Short Answer

Expert verified
The greatest speed is approximately 2.21 m/s.

Step by step solution

01

Convert Units

First, we need to convert the area from cm虏 to m虏. Since 1 m虏 = 10,000 cm虏, we convert 1.5 cm虏 to m虏. Therefore, 1.5 cm虏 = 1.5 脳 10鈦烩伌 m虏.
02

Calculate Maximum Force

We know that the force density (pressure) required to cause a rupture is 1.03 脳 10鈦� N/m虏. So, we calculate the maximum force that can be applied to the forehead using the pressure formula:\[ F = ext{Pressure} \times ext{Area} \]Substitute the given values:\[ F = 1.03 \times 10^8 \times 1.5 \times 10^{-4} \]\[ F = 1.545 \times 10^4 \text{ N} \]
03

Use Impulse-Momentum Theorem

Impulse is equal to the change in momentum, expressed as force times the time duration during which the force acts. The impulse-momentum theorem states:\[ F \times t = \Delta p \]Here, the mass (m) is 70.0 kg, the initial velocity is what we are looking for \( v \), and the final velocity is 0 m/s. The duration \( t \) is 10.0 ms or 0.010 s.
04

Calculate Change in Momentum

The change in momentum, \( \Delta p \), is:\[ \Delta p = m(v_f - v_i) = m(0 - v) = -mv \]Since \( F \times t = -mv \), we have:\[ 1.545 \times 10^4 \times 0.010 = 70.0 \times v \]\[ v = \frac{1.545 \times 10^4 \times 0.010}{70.0} \]
05

Solve for Maximum Speed

Calculate the maximum speed \( v \):\[ v = \frac{1.545 \times 10^4 \times 0.010}{70.0} \approx 2.207 \text{ m/s} \]
06

Conclusion: Maximum Safe Speed

The greatest speed with which the person can hit the wall without breaking any bone is approximately 2.21 m/s.

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Key Concepts

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

Force Density
Force density, often referred to as pressure, is a measure of force applied per unit area. It is expressed in units of N/m虏, where N stands for newtons, the unit of force, and m虏 is the unit of area.
In this exercise, the bone rupture threshold is given at a force density of 1.03 \(\times\) 10\(^8\) N/m虏. This parameter is crucial to know how much force the bone can tolerate before breaking.

Understanding force density can help design safer equipment or set realistic safety standards for various materials. It fundamentally describes how evenly a given force is distributed over a surface.
Pressure and Area Conversion
Pressure calculations often require accurate area measurements in appropriate units. Therefore, converting area units becomes critical. Pressure or force density calculations often use the area in squared meters (m虏), so any area provided in different units like squared centimeters (cm虏) must be converted.
In this problem, the area of contact with the forehead is provided as 1.5 cm虏. To convert it to m虏, we use the conversion factor: 1 m虏 = 10,000 cm虏. Thus, 1.5 cm虏 equals 1.5 \(\times\) 10\(^{-4}\) m虏.
  • This conversion is important as it ensures consistency in equations where N/m虏 is involved.
  • Unit consistency prevents calculation errors.
Keep these factors in mind when dealing with force density calculations.
Units Conversion
Units conversion is a critical skill in physics and engineering. Consistent units are necessary when performing calculations involving different physical quantities.
In this scenario, converting area units from cm虏 to m虏 is necessary to use the pressure given in N/m虏 accurately. Additionally, the time duration must be expressed in seconds instead of milliseconds to maintain consistency in the calculations. Here, 10.0 ms converts to 0.010 seconds.
  • Using consistent units ensures that final quantities like force and velocity are correctly computed.
  • Make sure to always double-check conversions to avoid inconsistencies.
Proper units play a pivotal role in producing accurate results in scientific calculations.
Momentum Calculation
Momentum, defined as the product of mass and velocity, is a vital concept in understanding motion dynamics. It is denoted by \( p \) and expressed as \( p = mv \), where \( m \) is mass and \( v \) is velocity.
The Impulse-Momentum Theorem relates the impulse acting on an object to its change in momentum. Impulse is the product of force and the duration it is applied, \( F \times t = \Delta p \).
In this exercise, we calculate the initial velocity of the person who hits a beam. Using the force determined from force density and the duration the force applies, the initial speed can be calculated before the person comes to a stop.
  • The momentum change, \( \Delta p \), is crucial to find how quickly the skater can hit the wall safely.
  • The impulse-momentum relationship showcases how forces affect motion over time.
Grasping this concept helps analyze collisions and movement accurately.

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