Posted: 9/2/2010 9:21:38 PM EDT
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Alright, this one may seem simple, but we didn't cover anything similar in the chapter or in the lecture, and I want to make sure that I'm doing this correctly... You have a two pulleys connected by a belt (say on a car engine). The first pulley has a 9" radius and the second has a 3" radius. If the pulley with the 9" radius rotates 25 degrees, then how many degrees does the smaller pulley rotate? So here is the way I did it: 9/3=3 25 degree * 3 = 75 degrees Am I doing it wrong? |
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I did it by looking at how much the belt is moving, since that's the only thing that ties the pulleys together. The circumfrence of the first pully is Pi * 2r = 56.52. The circumference of the second pulley is Pi*2r 18.4. The belt rotates 25 degrees, thus moving the belt some fraction of that circumference. (25/360) * 56.52 = 3.9 inches. What would it take to wrap that same 3.9 inches around the second pully? The fraction of the pully covered is 3.9 / the full circumference of 18.4, or 21%. 21% of 360 degrees is 75 degrees on the smaller pulley. So your original answer is right. |
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The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees.
That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. |
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Quoted: The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees. That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. Wat???? Circumference is Pi x Diameter. Area is Pi x Radius squared. Circumference of a 9" pulley is 28.26" The circumference of a 3" pulley is 9.42" But yes...The smaller pulley rotates 75 degrees. ETA: Never mind, I read diameter. |
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Quoted:
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The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees. That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. Wat???? Circumference of a 9" pulley is 28.26" ....(.0785" per degree) = 1.9625" in 25 degrees The circumference of a 3" pulley is 9.42" .... .0267" per degree The OP was tricky and specified radius instead of diameter. |
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Quoted: Quoted: Quoted: The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees. That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. Wat???? Circumference of a 9" pulley is 28.26" ....(.0785" per degree) = 1.9625" in 25 degrees The circumference of a 3" pulley is 9.42" .... .0267" per degree The OP was tricky and specified radius instead of diameter. Oh, I fucking failed hard on that one.  |
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Quoted: The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees. That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. Well, no one taught us the underlying principles, so they get the answer the best way I can figure it out. I thought that this was kind of odd, but at my school, the mathematics path is College Algebra ->Plane Trigonometry -> Analytic Geometry -> Pre-Calculus -> Calc 1 -> Calc 2. You have to transfer to a 4 year school to take Calc 3. |
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Let's go with a relation of Arc length and distance travelled.
Given: Pulley A has a radius of 9in, Rotates 25 degrees Pulley B has a radius of 3in Find: Rotation of pulley B, "theta_B" The arc length of a portion of the perimeter of a circle is given by the relation s=r*theta So we have pulleys A and B s_A = r_A * theta_A s_B = r_B * theta_B If the two pulleys are tied together with a belt, the linear distance travelled of a point on the belt is the arc length of pulley A rotated at some angle. per the problem statement, theta_A=25degrees Assuming that the belt is linearly rigid and does not stretch, then the arclength of pulley A (distance of belt travel) is equal to the arclength of pulley B. s_A = s_B or r_A * theta_A = r_B * theta_B Solving algebraically for theta_B theta_B = (r_A * theta_A ) / r_B ( one equation, one unknown) Now "plug'n'chug" (i.e. plug in given values, chug in the calculator) theta_B = ( 9(in) * 25 (deg) ) / 3 (in) (inches divide out leaving degrees) theta_B= 9/3 *25deg = 3*25 deg therefore theta_B= 75 degrees |
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Quoted: Let's go with a relation of Arc length and distance travelled. Given: Pulley A has a radius of 9in, Rotates 25 degrees Pulley B has a radius of 3in Find: Rotation of pulley B, "theta_B" The arc length of a portion of the perimeter of a circle is given by the relation s=r*theta So we have pulleys A and B s_A = r_A * theta_A s_B = r_B * theta_B If the two pulleys are tied together with a belt, the linear distance travelled of a point on the belt is the arc length of pulley A rotated at some angle. per the problem statement, theta_A=25degrees Assuming that the belt is linearly rigid and does not stretch, then the arclength of pulley A (distance of belt travel) is equal to the arclength of pulley B. s_A = s_B or r_A * theta_A = r_B * theta_B Solving algebraically for theta_B theta_B = (r_A * theta_A ) / r_B ( one equation, one unknown) Now "plug'n'chug" (i.e. plug in given values, chug in the calculator) theta_B = ( 9(in) * 25 (deg) ) / 3 (in) (inches divide out leaving degrees) theta_B= 9/3 *25deg = 3*25 deg therefore theta_B= 75 degrees The next unit is on linear velocity and angular velocity, but by this time nothing along these lines had been covered. Awesome info though. Thanks! |
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Quoted: Quoted: The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees. That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. Well, no one taught us the underlying principles, so they get the answer the best way I can figure it out. I thought that this was kind of odd, but at my school, the mathematics path is College Algebra ->Plane Trigonometry -> Analytic Geometry -> Pre-Calculus -> Calc 1 -> Calc 2. You have to transfer to a 4 year school to take Calc 3. Plane trig? Are you going to be an aeronautical engineer? |
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Quoted:
Quoted:
Let's go with a relation of Arc length and distance travelled. Given: Pulley A has a radius of 9in, Rotates 25 degrees Pulley B has a radius of 3in Find: Rotation of pulley B, "theta_B" The arc length of a portion of the perimeter of a circle is given by the relation s=r*theta So we have pulleys A and B s_A = r_A * theta_A s_B = r_B * theta_B If the two pulleys are tied together with a belt, the linear distance travelled of a point on the belt is the arc length of pulley A rotated at some angle. per the problem statement, theta_A=25degrees Assuming that the belt is linearly rigid and does not stretch, then the arclength of pulley A (distance of belt travel) is equal to the arclength of pulley B. s_A = s_B or r_A * theta_A = r_B * theta_B Solving algebraically for theta_B theta_B = (r_A * theta_A ) / r_B ( one equation, one unknown) Now "plug'n'chug" (i.e. plug in given values, chug in the calculator) theta_B = ( 9(in) * 25 (deg) ) / 3 (in) (inches divide out leaving degrees) theta_B= 9/3 *25deg = 3*25 deg therefore theta_B= 75 degrees The next unit is on linear velocity and angular velocity, but by this time nothing along these lines had been covered. Awesome info though. Thanks! Velocity is the derivative of distance with respect to time...linear velocity becomes V=r * omega s=r*theta derivative with respect to time: d/dt (s=r*theta) ds/dt = r * d theta/dt where V = ds/dt (rate of change of distance) and omega = d theta/dt (rate of change of angle)/ thus V=r * omega |
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Quoted: Quoted: Quoted: The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees. That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. Well, no one taught us the underlying principles, so they get the answer the best way I can figure it out. I thought that this was kind of odd, but at my school, the mathematics path is College Algebra ->Plane Trigonometry -> Analytic Geometry -> Pre-Calculus -> Calc 1 -> Calc 2. You have to transfer to a 4 year school to take Calc 3. Plane trig? Are you going to be an aeronautical engineer? Maybe! In mathematics, a plane is a two dimensional space. It means that we're studying Trigonometry in two dimension (height and width) rather than three dimensions. |
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Quoted: Quoted: Quoted: Quoted: The circumference of the 9" pulley is 56.52", and the circumference of the 3" pulley is 18.84". The 9" radius pulley is 0.157" per degree, and the 3" radius pulley is 0.0523" per degree. If the 9" radius pulley rotates by 25 degrees, it will turn 3.925". 3.925 divided by 0.0523 is 75. So the smaller pulley will rotate by 75 degrees. That's the fun and informative way to do it. The easy way (which you should only use once you understand the underlying concepts) is simply by ratios. 3 to 9 is 1 to 3, or 25 degrees to 75 degrees. Well, no one taught us the underlying principles, so they get the answer the best way I can figure it out. I thought that this was kind of odd, but at my school, the mathematics path is College Algebra ->Plane Trigonometry -> Analytic Geometry -> Pre-Calculus -> Calc 1 -> Calc 2. You have to transfer to a 4 year school to take Calc 3. Plane trig? Are you going to be an aeronautical engineer? Maybe! In mathematics, a plane is a two dimensional space. It means that we're studying Trigonometry in two dimension (height and width) rather than three dimensions. I know. It was a joke. |