course Phy 231 7/17 4:15 030. Rotational Motion Question: `q001. Note that this assignment contains 4 questions.
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Given Solution: The change of 20 degrees in 5 seconds implies a rate of change of 20 degrees / (5 seconds) = 4 deg / sec. We call this the angular velocity of the object, and we designate angular velocity by the symbol `omega. Self-critique (if necessary): OK Self-critique Assessment: OK ********************************************* Question: `q002. What is the average angular velocity of an object which rotates through an angle of 10 `pi radians in 2 seconds? Your solution: 10pi rad / 2s = 5pi rad/ second confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The average angular velocity is equal to the angular displacement divided by the time required for that displacement, in this case giving us `omega = `d`theta / `dt = 10 `pi radians / 2 seconds = 5 `pi rad/s. Self-critique (if necessary): OK Self-critique Assessment: OK ********************************************* Question: `q003. If an object begins with an angular velocity of 3 radians / sec and ends up 10 seconds later within angular velocity of 8 radians / sec, and if the angular velocity changes at a constant rate, then what is the average angular velocity of the object? In this case through how many radians this the object rotate and at what average rate does the angular velocity change? Your solution: Average angular velocity: 3 rad/s + 8 rad/s / 2 = 11 rad/s / 2 = 5.5 rad/s Average angular acceleration = 8rad/s – 3rad/s / 10s = 5 rad/s / 10s = .5 radians/second^2 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Starting at 3 rad/s and ending up at 8 rad/s, the average angular velocity would be expected to be greater than the minimum 3 rad/s and less than the maximum 8 rad/s. If the angular velocity changes at a constant rate, we would in fact expect the average angular velocity to lie halfway between 3 rad/s and 8 rad/s, at the average value (8 rad/s + 3 rad/s) / 2 = 5.5 rad/s. Moving at this average angular velocity for 10 sec the object would rotate through 5.5 rad/s * 10 s = 55 rad in 10 sec. The change in the angular velocity during this 10 seconds is (8 rad/s - 3 rad/s) = 5 rad/s; this change takes place in 10 seconds so that the average rate at which the angular velocity changes must be ( 5 rad / sec ) / (10 sec) = .5 rad/s^2. This is called the average angular acceleration. Angular acceleration is designated by the symbol lpha. Since the angular velocity in this example changes at a constant rate, the angular acceleration is constant and we therefore say that `alpha = `d `omega / `dt. Again in this case `d`omega is the 5 rad/sec change in the angular velocity. Self-critique (if necessary): OK Self-critique Assessment: OK ********************************************* Question: `q004. If an object starts out with angular velocity 14 rad/s and accelerates at a rate of 4 rad/s^2 for 5 seconds, then at what rate is the object rotating after the 5 seconds? Through how many radians will the object rotate during this time? Your solution: Vf = 14rad/s + 4 rad/s^2 * 5s Vf = 14rad/s + 20rad/s Vf = 34 radians/second vAve = 14rad/s + 34rad/s / 2 = 24 rad/s Radians = 24 rad/s * 5s = 120 radians confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Changing angular velocity at the rate of 4 rad/s^2 for 5 sec the angular velocity will change by (4 rad/s^2) (5s) = 20 rad/s. Since the angular velocity was already 14 rad/s at the beginning of this time period, it will be 14 rad/s + 20 rad/s = 34 rad/s at the end of the time period. The uniform rate of change of angular velocity implies that the average angular velocity is (14 rad/s + 34 rad/s) / 2 = 24 rad/s. An average angular velocity of 24 radians/second, in 5 seconds the object will rotate through an angle `d`theta = (24 rad/s) ( 5 sec) = 120 rad. Self-critique (if necessary): OK Self-critique Assessment: OK "