Problem 96
Question
A weight is moved upward through the use of a pulley 10 inches in radius. If the pulley is rotated counterclockwise through an angle of 45 ", approximate the height, in inches, that the weight will rise. Round your answer to two decimal places.
Step-by-Step Solution
Verified Answer
The weight will rise approximately 7.85 inches.
1Step 1: Convert the Angle to Radians
The angle is given in degrees, but the formula for arc length uses angles in radians. Therefore, the angle needs to be converted to radians. The formula used for this conversion is \( \theta_{rad} = \theta_{deg} * \frac{\pi}{180} \). Substituting \( \theta_{deg} = 45 \) into the equation gives \( \theta_{rad} = 45 * \frac{\pi}{180} = \frac{\pi}{4} \) rad.
2Step 2: Substituting Variables into Arc Length Formula
The formula for the arc length of a circle is \( s = r * \theta \). Substituting \( r = 10 \) inches and \( \theta = \frac{\pi}{4} \) rad obtained in step 1 into the formula gives \( s = 10 * \frac{\pi}{4} = \frac{10\pi}{4} = 2.5\pi \) inches.
3Step 3: Final Calculation and Rounding
To approximate the height to two decimal places, the value of \( \pi \) is approximated as 3.14. Therefore, substituting \( \pi = 3.14 \) into \( s = 2.5\pi \) gives \( s = 2.5 * 3.14 = 7.85 \) inches. This value is already rounded to two decimal places, so no further rounding is required.
Key Concepts
Radian MeasurePulley SystemCircle GeometryConverting Degrees to Radians
Radian Measure
The radian measure is a way of quantifying angles based on the radius of a circle. A radian is defined as the angle created at the center of a circle by an arc whose length is equal to the radius of the circle.
In other words, if you were to take the radius of a circle and trace it along the circumference, the angle subtended at the center of the circle by that arc would be one radian. This is an important concept in mathematics and physics because radians provide a 'natural' method of angle measurement, as opposed to degrees which are somewhat arbitrary (there are 360 degrees in a circle simply because ancient civilizations preferred the number 360).
There are exactly \(2\pi\) radians in a full circle. This is because the circumference of a circle is \(2\pi r\), where \(r\) is the radius, so when the arc length is equal to the circumference, the angle is a full turn, or \(2\pi\) radians.
In other words, if you were to take the radius of a circle and trace it along the circumference, the angle subtended at the center of the circle by that arc would be one radian. This is an important concept in mathematics and physics because radians provide a 'natural' method of angle measurement, as opposed to degrees which are somewhat arbitrary (there are 360 degrees in a circle simply because ancient civilizations preferred the number 360).
There are exactly \(2\pi\) radians in a full circle. This is because the circumference of a circle is \(2\pi r\), where \(r\) is the radius, so when the arc length is equal to the circumference, the angle is a full turn, or \(2\pi\) radians.
Pulley System
A pulley system is a simple machine that can lift, lower, or move a load with reduced effort. The basic pulley consists of a wheel with a grooved edge through which a rope or cable can run.
In practical applications, such as the exercise mentioned, a pulley's primary function is to change the direction of the force applied to move an object. That means a weight can be lifted straight up by pulling downward on the rope. The pulley system in this scenario is used to raise a weight through the rotation of the pulley, which is essentially converting rotational motion into linear motion of the weight.
The relationship between the rotation of the pulley and the actual linear displacement of the weight is directly proportional to the radius of the pulley. The larger the pulley, the higher the weight will rise for a given angle of rotation.
In practical applications, such as the exercise mentioned, a pulley's primary function is to change the direction of the force applied to move an object. That means a weight can be lifted straight up by pulling downward on the rope. The pulley system in this scenario is used to raise a weight through the rotation of the pulley, which is essentially converting rotational motion into linear motion of the weight.
The relationship between the rotation of the pulley and the actual linear displacement of the weight is directly proportional to the radius of the pulley. The larger the pulley, the higher the weight will rise for a given angle of rotation.
Circle Geometry
Circle geometry involves the properties and measurements associated with circles. Key concepts include the radius (the distance from the center of the circle to the edge), diameter (twice the radius, the distance across the circle through its center), and circumference (the total distance around the circle).
The arc length is another critical component of circle geometry; it is the measure of the distance along a section of the circumference. The arc length can be found by knowing the radius of the circle and the angle that subtends the arc. The central angle in radians and the radius form the basis of the arc length formula, \(s = r\theta\), where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the angle in radians.
This formula is crucial in applications involving circular motion or rotation, like gears, clock hands, and, as in this exercise, pulley systems.
The arc length is another critical component of circle geometry; it is the measure of the distance along a section of the circumference. The arc length can be found by knowing the radius of the circle and the angle that subtends the arc. The central angle in radians and the radius form the basis of the arc length formula, \(s = r\theta\), where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the angle in radians.
This formula is crucial in applications involving circular motion or rotation, like gears, clock hands, and, as in this exercise, pulley systems.
Converting Degrees to Radians
Converting degrees to radians is a necessary skill in mathematics, especially when dealing with trigonometry and geometry. Since most practical measurements are given in degrees, but many mathematical formulas (like the arc length formula) require the angle in radians, this conversion process is often applied.
To convert an angle given in degrees to radians, multiply the angle by \(\frac{\pi}{180}\). This factor comes from the relationship that there are 360 degrees in a circle, which is equivalent to \(2\pi\) radians — hence, 1 degree equals \(\frac{\pi}{180}\) radians.
Let's apply this to our 45-degree angle example: \[\theta_{rad} = \theta_{deg} \times \frac{\pi}{180} = 45 \times \frac{\pi}{180} = \frac{\pi}{4}\] radians.
Whenever you're working with circular measurements or trigonometry, remember to check if you need to perform this conversion to switch between degrees and radians effectively.
To convert an angle given in degrees to radians, multiply the angle by \(\frac{\pi}{180}\). This factor comes from the relationship that there are 360 degrees in a circle, which is equivalent to \(2\pi\) radians — hence, 1 degree equals \(\frac{\pi}{180}\) radians.
Let's apply this to our 45-degree angle example: \[\theta_{rad} = \theta_{deg} \times \frac{\pi}{180} = 45 \times \frac{\pi}{180} = \frac{\pi}{4}\] radians.
Whenever you're working with circular measurements or trigonometry, remember to check if you need to perform this conversion to switch between degrees and radians effectively.
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