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Two blocks (m 1 =2.5kg, m 2 =1.8kg) are hanging from a pulley as shown in the figure below. The moment of inertia through the axis of rotation passing through the center of the pulley is I=1.70kgm 2. The ropes are attached at two different distances from the center of the pulley.(R 1 =50cm and R 2 =20cm). Find the angular acceleration of the pulley system and the tensions T 1 and T 2. m1m1 m2m2 T2T2 T1T1 R2R2 R1R1 x y -a 1 a2a2

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Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Reflection

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1)What do we know about T 1 and T 2 ? A) B) C)

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Since the pulley is not massless (the pulley has a moment of inertia), the tension forces will not be the same, but at this point we do not know which is greater. Choice A Incorrect

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Choice B Correct Since the pulley is not massless (the pulley has a moment of inertia), the tension forces will not be the same. Also, the magnitudes of the accelerations of the blocks will not be the same, and the rotational motion of the pulley needs to be taken into consideration.

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Choice C Incorrect Since the pulley is not massless (the pulley has a moment of inertia), the tension forces will not be the same.

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2) What physics principle should we use to solve this problem? A) A) Newton’s 2nd Law B) B) Conservation of Energy C) C) Conservation of Momentum

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We will use both the translational and rotational form of Newton’s 2nd Law: Choice A Correct

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Since we are looking for the forces, torques, and accelerations of our system, the law of conservation of energy will not be helpful, because these quantities do not show up in the equations for energy explicitly. Choice B Incorrect

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Choice C Incorrect Since we are looking for the forces, torques, and accelerations of our system, the law of conservation of momentum will not be helpful, because these quantities do not show up in the equations for this law explicitly.

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3) Which of the following pairs of freebody diagrams correctly depicts all of the forces acting on masses 1 and 2? A) B) C) T1T1 T2T2 m1gm1gm2gm2g T1T1 m1gm1gm2gm2g T2T2 m1gm1gm2gm2g T1T1 T2T2 x y

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These diagrams correctly depict all of the forces acting on the two blocks. Choice A Correct

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Choice B Incorrect The pull of gravity (W=mg) acts in the negative y-direction on both blocks. Also, when a rope is attached to an object, the tension force is directed away from the object along the rope.

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Choice C Incorrect The are no forces acting in the positive or negative x-direction on either block.

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4) Which of the following expressions is a correct statement of Newton’s 2nd Law for the forces acting on mass one in the y-direction? A) B) C)

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The force of tension is not the only force acting upon block 1. Please check the freebody diagram and try again. Choice A Incorrect

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Choice B Correct Remember, the diagram with the original problem statement shows block one accelerating downwards. Using Newton’s 2nd Law and the freebody diagram for block 1, we find:

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Choice C Incorrect The block and pulley system is not in static equilibrium, therefore:

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5) Which of the following expressions is a correct statement of Newton’s 2nd Law for the forces acting on mass two in the y-direction? A) B) C)

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Choice A Incorrect The force of tension is not the only force acting upon block 2. Please check the freebody diagram and try again.

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Choice B Incorrect The block and pulley system is not in static equilibrium, therefore:

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Choice C Correct Using Newton’s 2nd Law and the freebody diagram for block 2, we find: Remember, the diagram with the original problem statement shows block two accelerating upwards.

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6) Which of the following expressions correctly relates acceleration (a) to angular acceleration ( )? A) B) C) R is the radius of a curve.

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Choice A Correct (Units in parenthesis)

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Choice B Incorrect Check the units. Acceleration should be in m/s 2 :

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Choice C Incorrect Check the units. Acceleration should be in m/s 2 :

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7) Using the rotational form of Newton’s 2nd Law, we get which of the following expressions? A) B) C)

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The forces from the blocks’ weights do not act upon the pulley directly, so they will not contribute to the net torque on the pulley. Choice A Incorrect

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Choice B Correct The two tension forces act in different directions on the pulley and cause rotation. They are multiplied by their associated lever arms, which are R 1 and R 2 in this case:

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Choice C Incorrect The forces from the blocks’ weights do not act upon the pulley directly, so they will not contribute to the net torque on the pulley. The tension forces act upon the pulley and cause it to rotate.

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8)Using the results of the previous questions, we get a set of three equations with three unknowns: Which expression is correct for the angular acceleration of the pulley ? A)A) B) C)B)C)

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Check your algebra and be careful with your signs. Choice A Incorrect

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Choice B Correct

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Choice C Incorrect Check your algebra and be careful with your signs.

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9) What is the numerical value for the angular acceleration of the pulley ? A) A) =7.0rad/s 2 B)B) =3.6rad/s 2 C)C) =2.6rad/s 2

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Choice A Incorrect Are you using the correct expression for ? Check your calculation. Correct Reasoning:

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Choice B Correct Notice: The pulley rotates in the counter clockwise direction Notice: The pulley rotates in the counter clockwise direction

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Choice C Incorrect Are you using the correct expression for ? Check your calculation. Correct Reasoning:

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10) What are the values for the tensions T 1 & T 2 ? A)A)T 1 =19.95NT 2 =18.95N B)B)T 1 =18.95NT 2 =19.95N C) C) T 1 =19.95NT 2 =0N

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Choice A Correct

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Choice B Incorrect Are you using the correct expressions for T 1 and T 2 ? Check your calculations.

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Choice C Incorrect The ropes connected to both blocks will be in tension. T 2 ≠0

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Reflection Questions: Which mass moves up and which mass moves down? What would the tensions and angular acceleration be if the positions of the blocks were switched?

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