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course phy201
`q001. Make a good sketch, as close to correct scale as possible using a free-hand sketch and without actually measuring, of a pendulum consisting of a string with a round mass on the end of the string, subject to the following conditions:The center of the mass is the origin of a coordinate system whose y axis is vertical and whose x axis is horizontal.
The length vector runs from the center of the mass (which is at the origin) to the point (-4 cm, 40 cm); at that point the pendulum string is attached to a rigid support.
Now on a separate xy coordinate plane, sketch the vector representing the tension exerted by the string on the pendulum mass.
Sketch the x and y projections of the tension vector.
Sketch the weight vector for the pendulum, assuming that the pendulum is in equilibrium with respect to the y direction (i.e., that the pendulum is not accelerating in the y direction), and that the only forces acting on the pendulum are its weight and string tension.
Note that the preceding conditions imply that the forces in the y direction must add up to zero. Hopefully you already knew that, but just in case you didn't think of it, check your sketch to make sure that it is so.
Estimate the magnitude of the x component of the tension vector as a percent of the magnitude of the weight vector. What is your estimate, based on your sketch?
40%
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Using the knowledge that the tension vector is parallel to the length vector, which runs from the origin to the point (40 cm, -4 cm), make your best improvement on your estimate. It is possible to arrive at the exact percent, but this would be expected only of University Physics students. Other students should be able to come pretty close, and some might well get the exact percent, but 'pretty close' is pretty good.
35%
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`q002. Continuing the preceding problem:
If the pendulum mass was subject to a net force which is equal to its weight, what would be its acceleration?
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The weight of the pendulum is m g, where m is its mass and g is the acceleration of gravity.
If the pendulum, whose mass is m, is subjected to force mg, its acceleration is
a = F / m = mg / m = g.
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If the pendulum mass was subject to a net force which is equal to half its weight, what would be its acceleration?
2x as much
A=f/m
A=1/.5
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Its weight is m g.
This answer needs to be modified.
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If the pendulum mass was subject to a net force which is equal to 7% of its weight, what would be its acceleration?
A=f/m
1/.07
A=.07/mass
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This also needs to be modified. 7% of its weight is .07 m g.
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According to your first estimate from the preceding problem, what would be the acceleration of the pendulum. (Hint: Its forces in the y direction are in equilibrium, so they add up to zero. The only force that's left is the x component of the tension).
A=f/.07(m)
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According to your second estimate, which was based on your knowledge of the components of the length vector, what would be the acceleration of the pendulum?
A=f/.35(m)
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This also needs to be modified.
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`q003. Again continuing the first problem:
What angle does the length vector make, as measured counterclockwise with respect to the positive x axis?
330 deg
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What angle does the tension vector make, as measured counterclockwise with respect to the positive x axis?
270 deg
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If the tension vector was equal in magnitude to the weight vector, what would be the magnitude of its vertical component as a percent of the weight?
95%
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I believe you said that the vertical component was 40% of the tension.
If the tension is equal to the weight and the vertical component is 40% of the tension, then what is the vertical component as a percent of the weight?
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Would the pendulum be in vertical equilibrium if this was the case?
no
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In this case, what is the magnitude of the y component of the net force as a percent of the weight?
Same, 100%
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If the y component of the tension was equal to the weight, then by what percent would the magnitude of the tension differ from the weight of the pendulum?
It wouldn’t differ
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The y component of the tension is only 40% of the tension (according to your estimate).
The y component of the tension is equal to the weight.
So the tension can't be equal to the weight.
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What is the x component of the tension as a percent of the weight?
100%
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This depends on the answer to the last question, which needs to be corrected before you correct this one.
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What is the magnitude of the net force on the pendulum mass as a percent of its weight?
3 x more
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What therefore is the acceleration of the pendulum?
A=f(3)/m
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The weight of the pendulum is m g.
So what is your answer to this question?
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University Physics students (who should think there is some small self-contradiction in the above analysis) and anyone else who also suspect this:
What is wrong with the assumption that the acceleration in the y direction is zero, how could this assumption be corrected, and how would it influence the analysis to make this correction? Note that a complete answer probably exceeds the scope of this course, but within the scope of the course it is at least possible to frame the question.
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`q004. If the same pendulum makes angle 30 degrees with the vertical, and if the y component of the tension is equal in magnitude to the weight of the mass, then what is the tension and what is its x component, both as a percent of its weight?
X component is = and tension is 80%
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If the pendulum is at 30 degrees then its x component is smaller than either the tension or its y component.
The y component of the tension is the weight.
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What seems inconsistent about the magnitude just obtained for the tension, in terms of the actual situation?
That the tension is almost equal to the weight
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If the tension was equal in magnitude to the weight, what would be its x component as a percent of the weight?
100%
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The x component of the tension is less than the tension.
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`q005. If the pendulum of question `q001, at the position given in that question, was moving at a speed of 50 cm / second, what would be its centripetal acceleration?
Centrip accel 30/360= 1/12
Acent (.5m/s)^2/.40m
.0125m/s^2
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How would this affect the tension?
It makes the tension increase because of the centripetal force
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What would the tension be as a percent of its weight?
30% f its weight
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How could this affect its acceleration along the arc?
It would slow it down
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`q006. University Physics:
A simple pendulum of length L consists of a mass m located at distance L from its axis of rotation. Its moment of inertia about that axis is therefore m L^2.
When the pendulum makes angle theta with vertical, the line of the gravitational force passes with distance L sin(theta) of the axis, so the moment-arm of the gravitational force is L sin(theta).
As you know, the gravitational force has magnitude m * g.
What therefore is the expression for the torque on the pendulum when its angle with vertical is theta?
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What is the angular acceleration of the system?
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Its angular velocity is the derivative omega = theta ' = dTheta / dt of the angular position of the pendulum.
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Its angular acceleration is the derivative alpha = omega ' = dOmega / dt of its angular velocity.
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Thus the angular acceleration is alpha = omega ' = theta ''.
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What therefore is the differential equation relating theta '' to theta for this pendulum? (unless you have nothing better to do, or have already completed a differential equations course, don't try to solve this equation)
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I'm not sure you have the picture of the pendulum at 30 degrees from the vertical. That pendulum would make angle 120 degrees relative to the positive x axis.
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Be sure to include the entire document, including my notes.
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