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Chapter 6: Problem 44

During an Olympic bobsled run, the Jamaican team makes a turn of radius 7.6 m at a speed of 96.6 km/h. What is their acceleration in terms of g?

Short Answer

Expert verified
The acceleration is approximately \(9.65g\).

Step by step solution

01

Convert Speed to Meters Per Second

The given speed is 96.6 km/h. To convert this to meters per second (m/s), use the conversion factor \(1\text{ km/h} = \frac{1}{3.6} \text{ m/s}\). So, \[ v = 96.6 \times \frac{1}{3.6} \approx 26.83 \text{ m/s}. \]
02

Use the Formula for Centripetal Acceleration

The formula for centripetal acceleration \(a_c\) is given by \(a_c = \frac{v^2}{r}\), where \(v\) is the speed in m/s and \(r\) is the radius in meters. Substituting the given values, \[ a_c = \frac{(26.83)^2}{7.6} \approx 94.68 \text{ m/s}^2. \]
03

Convert Acceleration to Terms of g

To express the acceleration in terms of \(g\), where \(g\) is the acceleration due to gravity (approximately \(9.81 \text{ m/s}^2\)), divide the calculated acceleration by \(g\). Thus, \[ a = \frac{94.68}{9.81} \approx 9.65g. \]

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Key Concepts

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

Physics Problems
Physics problems often challenge students by combining multiple concepts into a single question. In this exercise about bobsledding, we're dealing with the physics of motion in circular paths. Let's break this down:
  • Understanding motion and forces: In physics, motion involves how objects move in different paths. Here, the bobsled team is moving in a circular path, which introduces the concept of centripetal acceleration, essential in understanding rotational motion.
  • Grasping centripetal acceleration: Centripetal acceleration is critical as it keeps objects on a circular path. It's directed towards the center of the circle. Without this, the bobsled would move in a straight line, flying off the track.
  • Linking concepts: To solve this type of physics problem, students must connect centripetal acceleration with forces, as described by Newton's laws.
In these problems, it's also great practice to systematically work through equations and ensure each step logically follows from the last.
Unit Conversion
Unit conversion is a fundamental skill in physics. It ensures that all your measurements are in the correct units to solve problems accurately. Here, we perform a crucial conversion:
  • Speed conversion: The team's speed was initially given in kilometers per hour (km/h). But for physics calculations involving motion, the standard unit for speed is meters per second (m/s).
To convert from km/h to m/s, you divide by 3.6, since there are 3.6 km in a m/second. Therefore:\[v = 96.6 \times \frac{1}{3.6} \approx 26.83 \text{ m/s}.\]Having consistent units allows for accurate calculations when applying physics formulas and is vital in obtaining the correct results.
Newton's Laws of Motion
Newton's laws of motion lay the foundation for understanding the movement of objects. In our bobsled example, they help us conceptualize the forces involved:
  • First Law (Inertia): An object will remain in its state of motion unless acted upon by another force. For the bobsled, this force is centripetal.
  • Second Law (Acceleration): Acceleration occurs when a force acts on an object. The formula here is \(a = \frac{F}{m}\). In our example, we substitute the force with the centripetal force \(F = \frac{v^2}{r} \times m\). Solving for \(a\), we arrive at the centripetal acceleration formula, \(a_c = \frac{v^2}{r}\).
  • Third Law (Action-Reaction): Every action has an equal and opposite reaction. While not as directly applicable in this exercise, understanding that the ice pushes back on the sled helps us appreciate how these laws unify motion comprehension.
These laws help explain why the sled maintains its curved path and how to calculate the bobsled's transformation into an acceleration understood with respect to gravity ("g" units). Understanding these concepts ensures a strong grasp of fundamental physics.

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