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Chapter 5: Problem 58

The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is \(6.7 \mathrm{m},\) measured from its tip to the center of the circle. Find the ratio of the centripetal acceleration at the end of the blade to that which exists at a point located \(3.0 \mathrm{m}\) from the center of the circle.

Short Answer

Expert verified
The ratio of centripetal accelerations is approximately 0.448.

Step by step solution

01

Identify the Given Values

The length of the blade from the tip to the center is 6.7 m. We also have another point at 3.0 m from the center. We need to find the ratio of centripetal accelerations at these two distances.
02

Understand the Concept of Centripetal Acceleration

Centripetal acceleration is given by the formula \( a_c = \frac{v^2}{r} \), where \( v \) is the linear speed and \( r \) is the radius of the circle. Here, since the blade is rotating in a circle, the linear speed \( v \) is the same for any point along the blade.
03

Set Up the Ratio of Centripetal Accelerations

We want to find the ratio \( \frac{a_{c1}}{a_{c2}} \), where \( a_{c1} \) is the centripetal acceleration at 6.7 m and \( a_{c2} \) is the centripetal acceleration at 3.0 m. Using \( a_c = \frac{v^2}{r} \), we have: \[ a_{c1} = \frac{v^2}{6.7} \] \[ a_{c2} = \frac{v^2}{3.0} \]
04

Calculate the Ratio

Divide \( a_{c1} \) by \( a_{c2} \): \[ \frac{a_{c1}}{a_{c2}} = \frac{\frac{v^2}{6.7}}{\frac{v^2}{3.0}} = \frac{3.0}{6.7} \] After simplifying, we get: \[ \frac{3.0}{6.7} \approx 0.448 \]

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Key Concepts

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

Helicopter Blade Dynamics
Helicopter blade dynamics is the field that studies the motion and forces within helicopter rotor blades. The rotor blade is crucial because it offers the lift and thrust needed for flight. As such, understanding the dynamics involved is vital for pilots and engineers. Here's a brief breakdown to understand this better:
  • The rotor blades rotate around a central mast, creating airflow that generates lift.
  • Due to the rotation, the tips of the blades travel significantly faster than the parts closer to the center.
  • This affects the forces experienced by different points along the blade, influencing design decisions and operational limits.
When helicopter blades spin, they undergo rotational motion, thanks to their angular velocity. This motion causes variations in speed and, subsequently, different forces at various points along the blades. By understanding these dynamics, engineers can better design blades to ensure optimal performance and safety.
Rotational Motion
Rotational motion is a type of motion where an object spins or rotates around a fixed axis. This is seen extensively in helicopter blades, which rotate around a central point or mast. Here's more about this motion:
  • In rotational motion, every part of the object moves through a circular path.
  • Each point on the rotating object moves at a different speed, depending on its distance from the axis.
  • The closer a point is to the axis, the slower it moves; conversely, points farther from the center travel faster.
Rotational motion introduces concepts like angular velocities and centripetal forces, which are essential for analyzing the dynamics of rotating systems, such as helicopter blades.
Physics Problem Solving
Physics problem solving requires a structured approach to understand and tackle exercises. Here's a guide to solving the physics problem regarding the ratio of centripetal acceleration in helicopter blades:
  • **Identify what is given and what you need to find.** - For this problem, known distances are given along with the requirement to find a ratio of accelerations.
  • **Understand the relevant physics concepts involved.**- Here, centripetal acceleration is key, described by the formula \( a_c = \frac{v^2}{r} \) where \( r \) is the radius.
  • **Set up equations based on these concepts.**- Establish a formula for each known point and find their relation, as seen in the provided solution.
  • **Calculate and simplify.**- Finally, perform the calculations as in the solution, leading to the ratio \( \approx 0.448 \).
This step-by-step method helps in navigating complex physics problems, ensuring accuracy and understanding of the underlying principles.

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Most popular questions from this chapter

Multiple-Concept Example 7 reviews the concepts that play a role in this problem. Car A uses tires for which the coefficient of static friction is 1.1 on a particular unbanked curve. The maximum speed at which the car can negotiate this curve is \(25 \mathrm{m} / \mathrm{s}\). Car \(\mathrm{B}\) uses tires for which the coefficient of static friction is 0.85 on the same curve. What is the maximum speed at which car \(\mathrm{B}\) can negotiate the curve?

A satellite circles the earth in an orbit whose radius is twice the earth's radius. The earth's mass is \(5.98 \times 10^{24} \mathrm{kg}\), and its radius is \(6.38 \times 10^{6} \mathrm{m}\). What is the period of the satellite?

In a skating stunt known as crack-the-whip, a number of skaters hold hands and form a straight line. They try to skate so that the line rotates about the skater at one end, who acts as the pivot. The skater farthest out has a mass of \(80.0 \mathrm{kg}\) and is \(6.10 \mathrm{m}\) from the pivot. He is skating at a speed of \(6.80 \mathrm{m} / \mathrm{s}\). Determine the magnitude of the centripetal force that acts on him.

Ball A is attached to one end of a rigid massless rod, while an identical ball B is attached to the center of the rod, as shown in the figure. Each ball has a mass of \(m=0.50 \mathrm{kg}\), and the length of each half of the rod is \(L=0.40 \mathrm{m} .\) This arrangement is held by the empty end and is whirled around in a horizontal circle at a constant rate, so that each ball is in uniform circular motion. Ball A travels at a constant speed of \(v_{A}=5.0 \mathrm{m} / \mathrm{s} .\) Concepts: (i) How many tension forces contribute to the centripetal force that acts on ball A? (ii) How many tension forces contribute to the centripetal force that acts on ball B? (iii) Is the speed of ball A the same as that of ball B? Calculations: Find the tension in each half of the rod.

A jet flying at \(123 \mathrm{m} / \mathrm{s}\) banks to make a horizontal circular turn. The radius of the turn is \(3810 \mathrm{m},\) and the mass of the jet is \(2.00 \times 10^{5} \mathrm{kg}\) Calculate the magnitude of the necessary lifting force.
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