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Chapter 5: Problem 3

Presents a method for modeling this problem. The blade of a windshield wiper moves through an angle of \(90.0^{\circ}\) in \(0.40 \mathrm{~s}\). The tip of the blade moves on the arc of a circle that has a radius of \(0.45 \mathrm{~m}\). What is the magnitude of the centripetal acceleration of the tip of the blade?

Short Answer

Expert verified
The centripetal acceleration is approximately \( 6.92 \mathrm{~m/s^2} \).

Step by step solution

01

Understand the Problem

The problem gives us the angular distance covered by the wiper blade, the time taken, and the radius of the motion. We need to find the centripetal acceleration of the wiper blade's tip.
02

Convert Angle to Radians

Convert the given angular distance from degrees to radians. We know that \( 180^{\circ} = \pi \) radians. Therefore, \( 90.0^{\circ} = \frac{\pi}{2} \) radians.
03

Calculate Angular Speed

The angular speed \( \omega \) is calculated using the formula \( \omega = \frac{\theta}{t} \), where \( \theta \) is the angle in radians, and \( t \) is the time. Substituting, we get \( \omega = \frac{\pi/2}{0.40} = \frac{\pi}{0.80} \) rad/s.
04

Determine Centripetal Acceleration Formula

The centripetal acceleration \( a_c \) is given by \( a_c = \omega^2 r \), where \( \omega \) is the angular speed and \( r \) is the radius.
05

Calculate Centripetal Acceleration

Substitute \( \omega = \frac{\pi}{0.80} \) rad/s and \( r = 0.45 \mathrm{~m} \) into the centripetal acceleration formula: \[ a_c = \left(\frac{\pi}{0.80}\right)^2 \times 0.45 \]. Calculate the result to find \( a_c \approx 6.92 \mathrm{~m/s^2} \).

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Key Concepts

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

Centripetal Acceleration
When an object moves in a circular path, it experiences what is known as centripetal acceleration. This type of acceleration is always directed towards the center of the circle along which the object is moving. The formula for centripetal acceleration is expressed as \( a_c = \omega^2 r \), where \( \omega \) is the angular speed, and \( r \) is the radius of the circle. This formula highlights how both the speed of rotation and the size of the circle affect how sharply the path curves, thus influencing the required centripetal force.
Understanding this concept is important as it explains why, for instance, a faster spinning object requires a larger force to keep moving in a circular path.
Angular Speed
Angular speed, denoted by \( \omega \), describes how quickly an object rotates around a circular path. It is calculated by determining the angle covered per unit of time. We use the formula \( \omega = \frac{\theta}{t} \), where \( \theta \) is the angular displacement in radians, and \( t \) is time. Angular speed helps in identifying how fast a cycle repeats itself over time.
Moreover, angular speed is different from linear speed. While linear speed tells us the straight-line speed, angular speed is specific to rotations around a circle, making it crucial for problems involving circular motion.
Radians Conversion
Degrees and radians are two units to measure angles. Converting between degrees and radians is a necessary skill in physics, especially for problems involving circular motion. For conversion, remember that \( 180^{\circ} = \pi \) radians. Hence, to convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \). In the given problem, the windshield wiper blade covers \( 90.0^{\circ} \), which is converted to \( \frac{\pi}{2} \) radians.
This conversion is essential when dealing with trigonometric functions or any physics formula that requires radians. Practicing these conversions will improve problem solving skills in physics.
Problem Solving Steps
Breaking down problems into steps is crucial in physics. Let's summarize the steps from our windshield wiper problem:
  • Understand the Problem: Identify what is given and what needs to be found. Here, we need the centripetal acceleration of the wiper's tip.
  • Convert Measurements: Convert angles to radians if necessary, which was done from \( 90.0^{\circ} \) to \( \frac{\pi}{2} \) radians.
  • Calculate Angular Speed: Use \( \omega = \frac{\theta}{t} \) with our converted angle.
  • Apply Formulas: Insert variables into the appropriate formulas, like \( a_c = \omega^2 r \), to find the solution.
Each step helps in structuring the problem-solving process and ensures that all needed data is correctly applied, leading to a successful solution.

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Most popular questions from this chapter

A curve of radius \(120 \mathrm{~m}\) is banked at an angle of \(18^{\circ} .\) At what speed can it be negotiated under icy conditions where friction is negligible?

A rigid massless rod is rotated about one end in a horizontal circle. There is a mass \(m_{1}\) attached to the center of the rod and a mass \(m_{2}\) attached to the outer end of the rod. The inner section of the rod sustains three times as much tension as the outer section. Find the ratio \(m_{2} / m_{1}\)

There is a clever kitchen gadget for drying lettuce leaves after you wash them. It consists of a cylindrical container mounted so that it can be rotated about its axis by turning a hand crank. The outer wall of the cylinder is perforated with small holes. You put the wet leaves in the container and turn the crank to spin off the water. The radius of the container is 12 \(\mathrm{cm}\). When the cylinder is rotating at 2.0 revolutions per second, what is the magnitude of the centripetal acceleration at the outer wall?

Pilots of high-performance fighter planes can be subjected to large centripetal accelerations during high-speed turns. Because of these accelerations, the pilots are subjected to forces that can be much greater than their body weight, leading to an accumulation of blood in the abdomen and legs. As a result, the brain becomes starved for blood, and the pilot can lose consciousness ("black out"). The pilots wear "anti-G suits" to help keep the blood from draining out of the brain. To appreciate the forces that a fighter pilot must endure, consider the magnitude \(F_{\mathrm{N}}\) of the normal force that the pilot's seat exerts on him at the bottom of a dive. The magnitude of the pilot's weight is \(W\). The plane is traveling at \(230 \mathrm{~m} / \mathrm{s}\) on a vertical circle of radius \(690 \mathrm{~m}\). Determine the ratio \(F_{\mathrm{N}} / W\). For comparison, note that black-out can occur for values of \(F_{\mathrm{N}} / W\) as small as 2 if the pilot is not wearing an anti-G suit.

How long does it take a plane, traveling at a constant speed of \(110 \mathrm{~m} / \mathrm{s}\), to fly once around a circle whose radius is \(2850 \mathrm{~m} ?\)
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