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Part A Consider another special case in which the inclined plane is vertical (0 = π/2). In this case, for what value of my would the acceleration of the two blocks be equal to zero? Express your answer in terms of some or all of the variables m2 and g. Π| ΑΣΦ 1 m₁ = Submit Request Answer ?

Part A Consider another special case in which the inclined plane is vertical (0 = π/2). In this case, for what value of my would the acceleration of the two blocks be equal to zero? Express your answer in terms of some or all of the variables m2 and g. Π| ΑΣΦ 1 m₁ = Submit Request Answer ?

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Learning Goal: Once you have decided to solve a problem using Newton's 2nd law, there are steps that will lead you to a solution. One such prescription is the following: • Visualize the problem and identify special cases. Isolate each body and draw the forces acting on it. Choose a coordinate system for each body. Apply Newton's 2nd law to each body. • Write equations for the constraints and other given information. • Solve the resulting equations symbolically. ● Check that your answer has the correct dimensions and satisfies special cases. ● ● ● • If numbers are given in the problem, plug them in and check that the answer makes sense. • Think about generalizations or simplifications of the problem. As an example, we will apply this procedure to find the acceleration of a block of mass m2 that is pulled up a frictionless plane inclined at angle with respect to the horizontal by a massless string that passes over a massless, frictionless pulley to a block of mass m₁ that is hanging vertically. (Figure 1) Figure block 2 block 1 1 of 2
Transcribed Image Text:Learning Goal: Once you have decided to solve a problem using Newton's 2nd law, there are steps that will lead you to a solution. One such prescription is the following: • Visualize the problem and identify special cases. Isolate each body and draw the forces acting on it. Choose a coordinate system for each body. Apply Newton's 2nd law to each body. • Write equations for the constraints and other given information. • Solve the resulting equations symbolically. ● Check that your answer has the correct dimensions and satisfies special cases. ● ● ● • If numbers are given in the problem, plug them in and check that the answer makes sense. • Think about generalizations or simplifications of the problem. As an example, we will apply this procedure to find the acceleration of a block of mass m2 that is pulled up a frictionless plane inclined at angle with respect to the horizontal by a massless string that passes over a massless, frictionless pulley to a block of mass m₁ that is hanging vertically. (Figure 1) Figure block 2 block 1 1 of 2
Visualize the problem and identify special cases First examine the problem by drawing a picture and visualizing the motion. Apply Newton's 2nd law, F = mã, to each body in your mind. Don't worry about which quantities are given. Think about the forces on each body: How are these consistent with the direction of the acceleration for that body? Can you think of any special cases that you can solve quickly now and use to test your understanding later? One special case in this problem is if m2 = 0, in which case block 1 would simply fall freely under the acceleration of gravity: a₁ = −gj. Part A Consider another special case in which the inclined plane is vertical (0 = π/2). In this case, for what value of m₁ would the acceleration of the two blocks be equal to zero? Express your answer in terms of some or all of the variables m2 and g. - ΓΙ ΑΣΦ m1 = Submit Request Answer ?
Transcribed Image Text:Visualize the problem and identify special cases First examine the problem by drawing a picture and visualizing the motion. Apply Newton's 2nd law, F = mã, to each body in your mind. Don't worry about which quantities are given. Think about the forces on each body: How are these consistent with the direction of the acceleration for that body? Can you think of any special cases that you can solve quickly now and use to test your understanding later? One special case in this problem is if m2 = 0, in which case block 1 would simply fall freely under the acceleration of gravity: a₁ = −gj. Part A Consider another special case in which the inclined plane is vertical (0 = π/2). In this case, for what value of m₁ would the acceleration of the two blocks be equal to zero? Express your answer in terms of some or all of the variables m2 and g. - ΓΙ ΑΣΦ m1 = Submit Request Answer ?
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**Part B** Which of the four drawings is a correct force diagram for this problem? ([Figure 2](#)) - a - b - c - d [Submit] [Request Answer] --- **Choose a coordinate system for each body** Newton's 2nd law, \(\sum \vec{F} = m\vec{a}\), is a vector equation. To add or subtract vectors, it is often easiest to decompose each vector into components. Whereas a particular set of vector components is only valid in a particular coordinate system, the vector equality holds in any coordinate system, giving you freedom to pick a coordinate system that most simplifies the equations that result from the component equations. It’s generally best to pick a coordinate system where the acceleration of the system lies directly on one of the coordinate axes. If there is no acceleration, then pick a coordinate system with as many unknowns as possible along the coordinate axes. Vectors that lie along the axes appear in only one of the equations for each component, rather than in two equations with trigonometric prefactors. Note that it is sometimes advantageous to use different coordinate systems for each body in the problem. In this problem, you should use Cartesian coordinates and your axes should be stationary with respect to the inclined plane. --- **Part C** Given the criteria just described, what orientation of the coordinate axes would be best to use in this problem? In the answer options, "tilted" means with the x axis oriented parallel to the plane (i.e., at angle \(\theta\) to the horizontal), and "level" means with the x axis horizontal. - tilted for both block 1 and block 2 - tilted for block 1 and level for block 2 - level for block 1 and tilted for block 2 - level for both block 1 and block 2 [Submit] [Request Answer]
Transcribed Image Text:**Part B** Which of the four drawings is a correct force diagram for this problem? ([Figure 2](#)) - a - b - c - d [Submit] [Request Answer] --- **Choose a coordinate system for each body** Newton's 2nd law, \(\sum \vec{F} = m\vec{a}\), is a vector equation. To add or subtract vectors, it is often easiest to decompose each vector into components. Whereas a particular set of vector components is only valid in a particular coordinate system, the vector equality holds in any coordinate system, giving you freedom to pick a coordinate system that most simplifies the equations that result from the component equations. It’s generally best to pick a coordinate system where the acceleration of the system lies directly on one of the coordinate axes. If there is no acceleration, then pick a coordinate system with as many unknowns as possible along the coordinate axes. Vectors that lie along the axes appear in only one of the equations for each component, rather than in two equations with trigonometric prefactors. Note that it is sometimes advantageous to use different coordinate systems for each body in the problem. In this problem, you should use Cartesian coordinates and your axes should be stationary with respect to the inclined plane. --- **Part C** Given the criteria just described, what orientation of the coordinate axes would be best to use in this problem? In the answer options, "tilted" means with the x axis oriented parallel to the plane (i.e., at angle \(\theta\) to the horizontal), and "level" means with the x axis horizontal. - tilted for both block 1 and block 2 - tilted for block 1 and level for block 2 - level for block 1 and tilted for block 2 - level for both block 1 and block 2 [Submit] [Request Answer]
The image presents four diagrams labeled (a), (b), (c), and (d), illustrating the forces acting on two blocks in different scenarios. Each diagram includes vectors representing different forces. ### Diagram (a): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( m_2a \) (mass times acceleration). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - An upward force vector labeled \( m_1a \). - A downward force vector labeled \( m_1g \) (weight due to gravity). ### Diagram (b): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( T_2 \) (tension). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - An upward force vector labeled \( T_1 \) (tension). - A downward force vector labeled \( m_1g \) (weight due to gravity). ### Diagram (c): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( m_2a \) (mass times acceleration). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - Two downward force vectors labeled \( m_1g \) and \( m_1a \). ### Diagram (d): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( T_2 \) (tension). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - An upward force vector labeled \( T_1 \) (tension). - Two downward force vectors labeled \( m_1g \) and \( m_1a \). These diagrams demonstrate different force interactions in scenarios involving normal forces, tension, gravity, and acceleration.
Transcribed Image Text:The image presents four diagrams labeled (a), (b), (c), and (d), illustrating the forces acting on two blocks in different scenarios. Each diagram includes vectors representing different forces. ### Diagram (a): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( m_2a \) (mass times acceleration). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - An upward force vector labeled \( m_1a \). - A downward force vector labeled \( m_1g \) (weight due to gravity). ### Diagram (b): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( T_2 \) (tension). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - An upward force vector labeled \( T_1 \) (tension). - A downward force vector labeled \( m_1g \) (weight due to gravity). ### Diagram (c): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( m_2a \) (mass times acceleration). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - Two downward force vectors labeled \( m_1g \) and \( m_1a \). ### Diagram (d): - **Block 2**: - An upward force vector labeled \( N \) (normal force). - A leftward force vector labeled \( T_2 \) (tension). - A downward force vector labeled \( m_2g \) (weight due to gravity). - **Block 1**: - An upward force vector labeled \( T_1 \) (tension). - Two downward force vectors labeled \( m_1g \) and \( m_1a \). These diagrams demonstrate different force interactions in scenarios involving normal forces, tension, gravity, and acceleration.
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