Ch 05 HW

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3/24/23, 2:08 PM Ch 05 HW https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10460269 1/53 Ch 05 HW Due: 11:59pm on Tuesday, March 21, 2023 You will receive no credit for items you complete after the assignment is due. Grading Policy Boat Statics A boat owner pulls her boat into the dock shown, where there are six bollards to which to tie the boat. She has three ropes. She can tie the boat from the boat's center (A) to any of the bollards (B through G) along the dotted arrows shown. Suppose the owner has tied three ropes: one rope runs to A from B, another to A from D, and a final rope from A to F. The ropes are tied such that F AB = F AD . The following notation is used in this problem: When a question refers to, for example, → F AB F_AB_vec , this quantity is taken to mean the force acting on the boat due to the rope running to A from B, while F AB F_AB is the magnitude of that force. Part A What is the magnitude of the force provided by the third rope, in terms of θ theta ? Hint 1. Find the forces in the x direction What is the component of → F AB F_AB_vec in the x direction (call it simply F x F_x ), in terms of F AB F_AB and θ theta ? Positive x is to the right in the diagram. ANSWER: Hint 2. Using algebra and trigonometry Recalling that F AB = F AD , you can find the x component of the net force due to → F AB F_AB_vec and → F AD F_AD_vec . How does this relate to the force provided by the third rope? ANSWER: Correct F x = − F AB cos( θ ) F AB cos( θ ) 2 F AB cos( θ ) 2 F AB sin( θ ) F AB sin( θ )

3/24/23, 2:08 PM Ch 05 HW https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10460269 2/53 Atwood Machine Special Cases shows an Atwood machine that consists of two blocks (of masses m 1 and m 2 ) tied together with a massless rope that passes over a fixed, perfect (massless and frictionless) pulley. In this problem you'll investigate some special cases where physical variables describing the Atwood machine take on limiting values. Often, examining special cases will simplify a problem, so that the solution may be found from inspection or from the results of a problem you've already seen. For all parts of this problem, take upward to be the positive direction and take the gravitational constant, g , to be positive. Part A Consider the case where m 1 and m 2 are both nonzero, and m 2 > m 1 . Let T 1 be the magnitude of the tension in the rope connected to the block of mass m 1 , and let T 2 be the magnitude of the tension in the rope connected to the block of mass m 2 . Which of the following statements is true? ANSWER: Correct Part B Now, consider the special case where the block of mass m 1 is not present. Find the magnitude, T , of the tension in the rope. Try to do this without equations; instead, think about the physical consequences. Hint 1. How to approach the problem If the block of mass m 1 is not present, and the rope connecting the two blocks is massless, will the motion of the block of mass m 2 be any different from free fall? Hint 2. Which physical law to use Use Newton's 2nd law on the block of mass m 2 . ANSWER: Correct Part C T 1 is always equal to T 2 . T 2 is greater than T 1 by an amount independent of velocity. T 2 is greater than T 1 but the difference decreases as the blocks increase in velocity. There is not enough information to determine the relationship between T 1 and T 2 . T = 0

3/24/23, 2:08 PM Ch 05 HW https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10460269 3/53 For the same special case (the block of mass m 1 not present), what is the acceleration of the block of mass m 2 ? Express your answer in terms of g , and remember that an upward acceleration should be positive. ANSWER: Correct Part D Next, consider the special case where only the block of mass m 1 is present. Find the magnitude, T , of the tension in the rope. ANSWER: Correct Part E For the same special case (the block of mass m 2 not present) what is the acceleration of the end of the rope where the block of mass m 2 would have been attached? Express your answer in terms of g , and remember that an upward acceleration should be positive. ANSWER: Correct Part F Next, consider the special case m 1 = m 2 = m . What is the magnitude of the tension in the rope connecting the two blocks? Use the variable m in your answer instead of m 1 or m 2 . ANSWER: Correct Part G For the same special case ( m 1 = m 2 = m ), what is the acceleration of the block of mass m 2 ? ANSWER: Correct Part H Finally, suppose m 1 → ∞ , while m 2 remains finite. What value does the the magnitude of the tension approach? a 2 = -9.80 T = 0 a 2 = 9.80 T = mg a 2 = 0

3/24/23, 2:08 PM Ch 05 HW https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10460269 4/53 Hint 1. Acceleration of block of mass m 1 As m 1 becomes large, the finite tension T will have a neglible effect on the acceleration, a 1 . If you ignore T , you can pretend the rope is gone without changing your results for a 1 . As m 1 → ∞ , what value does a 1 approach? ANSWER: Hint 2. Acceleration of block of mass m 2 As m 1 → ∞ , what value will the acceleration of the block of mass m 2 approach? ANSWER: Hint 3. Net force on block of mass m 2 What is the magnitude F net of the net force on the block of mass m 2 . Express your answer in terms of T , m 2 , g , and any other given quantities. Take the upward direction to be positive. ANSWER: ANSWER: Correct Imagining what would happen if one or more of the variables approached infinity is often a good way to investigate the behavior of a system. Two Masses, a Pulley, and an Inclined Plane Block 1, of mass m 1 m_1 , is connected over an ideal (massless and frictionless) pulley to block 2, of mass m 2 m_2 , as shown. Assume that the blocks accelerate as shown with an acceleration of magnitude a a and that the coefficient of kinetic friction between block 2 and the plane is μ mu . Part A Find the ratio of the masses m 1 / m 2 . a 1 = -9.80 a 2 = 9.80 F net = T − m 2 g T = 2 m 2 g

3/24/23, 2:08 PM Ch 05 HW https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10460269 5/53 Express your answer in terms of some or all of the variables a a , μ mu , and θ theta , as well as the magnitude of the free-fall acceleration g g . Hint 1. Draw a free-body diagram Which figure depicts the correct free-body diagrams for the blocks in this problem? Figure a) Figure b) Figure c) ANSWER: a b c

3/24/23, 2:08 PM Ch 05 HW https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10460269 6/53 Hint 2. Apply Newton's 2nd law to block 2 in the direction parallel to the incline What is Newton's 2nd law for block 2 in the direction parallel to the incline? (Assume the positive direction is going up the incline.) Express your answer in terms of m 2 m_2 , T T , f f , and θ theta , as well as the magnitude of the free-fall acceleration g g . ANSWER: Hint 3. Find an expression for the friction force What is the magnitude f f of the friction force acting on block 2? ANSWER: Hint 4. Find the normal force By applying Newton's 2nd law to block 2 in the direction perpendicular to the incline determine the magnitude of the normal force n n . Express your answer in terms of m 2 m_2 and θ theta , as well as the magnitude of the free-fall acceleration g g . ANSWER: Hint 5. Apply Newton's 2nd law to block 1 in the vertical direction Write an expression for Newton's 2nd law in the vertical direction for block 1. Take the positive direction to point downward. Express your answer in terms of the variables m 1 m_1 and/or T T , as well as the magnitude of the free-fall acceleration g g . ANSWER: Hint 6. Solve for the unknown tension T T Using your result from the previous hint, express T T in terms of g g , a a , and m 1 m_1 . ANSWER: Hint 7. Putting it all together By applying Newton's law to both block 1 and block 2, as you did in Hints 2 and 4, you found two equations where the masses m 1 m_1 , m 2 m_2 , the tension T T , and the acceleration a a , all appear, along with g g and θ theta . (Note that the friction force can be expressed in terms of the normal force, which, in turn, can be written as m 2 g cos( θ ) , as you found in Hint 3.) Choose one of the two equation and solve for T T ; substitute this result into the other equation. You will then have an equation with factors of m 1 m_1 and m 2 m_2 . You can then deduce the ratio. ANSWER: m 2 a = T − f − m 2 g sin( θ ) f = n / μ f = μ / n f = μn → f = μ → n n n = m 2 g cos( θ ) m 1 a = m 1 g − T T T = m 1 ( g − a ) m 1 / m 2 = a + ( μ cos θ + sin θ ) g g − a

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