Homework #19 (due Friday 5/6)

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Brianna Neal
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1 Homework #19 (due Friday 5/6) Physics ID number Group Letter One issue that people often have trouble with at this point is distinguishing between tangential acceleration and centripetal acceleration for something moving in a circular path. The centripetal acceleration, =, is the acceleration inward towards the center of the circle that something has, to keep it moving along the circular path. An object moving in a circle will have this acceleration regardless of whether the object s speed is changing. The tangential acceleration, at, is an acceleration tangent to the circular path something can have if it s speed is changing. For example, a car traveling around a circular path at constant speed would only have the centripetal acceleration, no tangential acceleration. If the car s speed starts changing, then it would also have a tangential acceleration, at. This tangential acceleration is the one that is related to the angular acceleration,, by a= r. Note that both of the a accelerations, centripetal and tangential, are measured in m/s 2. The angular acceleration is the one measured in rad/s (10 points) a. A car is sitting still on a circular racetrack, with a radius of 100 meters. It starts accelerating at a rate of 6 m/s 2 at t=0. This is its tangential acceleration the acceleration it has tangent to the circle, measured in m/s 2. What is the car s angular acceleration? b. At t=5 seconds, how far has the car gone in meters? c. At t=5 seconds, how far has the car gone in radians? d. At t=5 seconds, how fast is the car going in m /s? e. At t=5 seconds, how fast is the car going in rad /s?

2 f. At t=5 seconds, what is the car s centripetal acceleration? g. On the picture to the right, the car is shown at t=0, with its acceleration vector. At that moment, it is only accelerating tangent to the circle. Since its speed is zero at that moment, it has no centripetal acceleration. Draw the car where it is located at t=5 seconds. Also draw its tangential acceleration (which should still be 6 m/s 2 ) and its centripetal acceleration. h. Draw the car s net acceleration, and calculate its magnitude (this is the combination of the centripetal and tangential acceleration). The point of this problem is to hopefully clarify the difference between tangential and centripetal acceleration. t=0 a t=6 m/s 2 The new one that has been introduced is tangential acceleration, which we should write as at, and this is the one that equals r.

3 2. (10 points) When doing Newton s 2 nd Law stuff, we analyzed many systems involving things connected by ropes to pulleys. The only thing that has changed now is that we are no longer assuming the pulleys are massless that was a simplification that we were making, which is valid as long as the mass of the pulley is very small compared to the other masses involved. But that isn t always the case. If the pulley mass is not negligible, that means that the tensions in the rope on the two sides of the pulley will not be the same, since a net torque is necessary to make the pulley accelerate (unless its mass is zero), by =. So we have to call the tension on each side of the pulley something different (i.e. T1 and T2), which means there are two unknown tensions. So we need another equation which comes from =. Other than that, the process is the same draw FBDs (of objects moving linearly and something rotating, like a pulley), write out Newton s 2 nd Law (linear for things accelerating linearly, angular form for things spinning), and then do the algebra. Consider this system, with friction between the 2.0 kg block and the incline. a. Draw FBDs of both blocks and also both pulleys. b. How many unknowns do you have? (you know the masses of both blocks, and the coefficient of friction) List your unknowns. You need to have as many equations as unknowns to be able to solve for them all. If you have different accelerations, indicate how they relate to each other.

4 c. Write out Newton s 2 nd Law in linear forms for each object that accelerates linearly, and in angular form for each object that accelerates angularly. d. Do you have as many equations as unknowns? Don t bother working through the algebra. Just make sure you are comfortable doing the FBDs, writing out your Newton s 2 nd Law equations in both forms, how you can relate any accelerations to each other, and then how you would do the algebra. In a situation like this the algebra can get messy and it s not our primary goal.

5 3. (10 points) So far when analyzing pendulums, we ve assumed the rope (or chain) has no mass, and all the mass is at the end of the pendulum we treated it as a point mass. That allowed us to analyze things with our linear approach, since all the mass was at one spot, all moving together. If the mass is not all at one spot, we can t do that. For example, consider a rigid pendulum made by connecting the end of a rod to an axle, as shown below. The rod has a length L and mass m. The mass of the rod though is distributed uniformly along it, rather than completely at the end so we can t treat it as a point mass. Our goal is to figure out how fast the rod is moving when it gets to the bottom, assuming it started out horizontal initially. a. With a regular pendulum, we can do that with conservation of energy, with the pendulum bob having linear kinetic energy (since the mass is purely at one spot). Now, we can still use conservation of energy, but the rod gains rotational kinetic energy as it spins. A long thin rod has a rotational inertia (moment of inertia) of a rod rotating around its end is. Use conservation of energy to determine the angular velocity of the rod at the bottom. pivot rod b. What is the tangential velocity (linear speed, vt) of the very end of the rod? c. If this were a simple pendulum, with the mass entirely at the end, what would its speed be when it s at the bottom? Is the tip of the rod going faster or slower than a point-mass pendulum?

6 4. (10 points) This last part has nothing to do with rotational stuff. As part of some of our education research, we want to see how students are thinking about certain things by the end of the semester. To that end, please complete the quick survey at this web address: In non-link form, in case it s easier to see:

Webreview Torque and Rotation Practice Test

Webreview Torque and Rotation Practice Test Please do not write on test. ID A Webreview - 8.2 Torque and Rotation Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A 0.30-m-radius automobile