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For this problem, take a look at Figure 2 below. A disk with uniformly distributed mass m, radius R, and center of mass at point O is connected to a combination of springs at point P, which are then connected to a fixed wall. The disk rolls without slipping at point Q along an inclined plane that is at an angle a from the horizontal. Gravity acts in the vertical direction (towards the bottom of the page). ₁ is the linear coordinate of the point O along the inclined plane. The positive direction of ri is as shown. When the springs are undeflected, ₁ = 0. An angle , about the instant center of rotation, is shown. You may assume that the motion (and therefore angle 0) is small. puny m Massless structure between springs Figure 2: System schematic. Your tasks: A Draw the FBD for the disk. Don't forget the forces at point Q B Derive the equation of motion with as the dynamic variable. Be sure to put it in input-output standard form (inputs and constant forces on the right, things related to the dynamic variable and its derivatives on the left). You must also write out the mass moment of inertia of the system about the point Q, IQ, in terms of the given variables. C Convert the equation of motion so that it is in terms of a1, the linear coordinate associated with the center of the disk. Again, be sure it is in input-output standard form.

For this problem, take a look at Figure 2 below. A disk with uniformly distributed mass m, radius R, and center of mass at point O is connected to a combination of springs at point P, which are then connected to a fixed wall. The disk rolls without slipping at point Q along an inclined plane that is at an angle a from the horizontal. Gravity acts in the vertical direction (towards the bottom of the page). ₁ is the linear coordinate of the point O along the inclined plane. The positive direction of ri is as shown. When the springs are undeflected, ₁ = 0. An angle , about the instant center of rotation, is shown. You may assume that the motion (and therefore angle 0) is small. puny m Massless structure between springs Figure 2: System schematic. Your tasks: A Draw the FBD for the disk. Don't forget the forces at point Q B Derive the equation of motion with as the dynamic variable. Be sure to put it in input-output standard form (inputs and constant forces on the right, things related to the dynamic variable and its derivatives on the left). You must also write out the mass moment of inertia of the system about the point Q, IQ, in terms of the given variables. C Convert the equation of motion so that it is in terms of a1, the linear coordinate associated with the center of the disk. Again, be sure it is in input-output standard form.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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**For this problem,** refer to Figure 2 below. A disk with uniformly distributed mass \( m \), radius \( R \), and center of mass at point \( O \) is connected to a combination of springs at point \( P \), which are then connected to a fixed wall. The disk rolls without slipping at point \( Q \) along an inclined plane that is at an angle \( \alpha \) from the horizontal. Gravity acts in the vertical direction (towards the bottom of the page). \( x_1 \) is the linear coordinate of the point \( O \) along the inclined plane. The positive direction of \( x_1 \) is as shown. When the springs are undeflected, \( x_1 = 0 \). An angle \( \theta \), about the instant center of rotation, is shown. You may assume that the motion (and therefore angle \( \theta \)) is small. **Figure Description:** - A disk is shown rolling along an inclined plane with an angle \( \alpha \). - The disk has a center of mass at point \( O \) and is connected to springs at point \( P \). - The springs are connected to each other and then to a fixed wall, labeled as a “Massless structure between springs.” - An angle \( \theta \) is depicted between a line perpendicular to the incline and a line connecting the point of contact and the center of the disk. - Other key points and dimensions include the radius \( R \), and the linear coordinate \( x_1 \). **Your Tasks:** **A**: Draw the Free Body Diagram (FBD) for the disk. Don’t forget the forces at point \( Q \). **B**: Derive the equation of motion with \( \theta \) as the dynamic variable. Be sure to put it in input-output standard form (inputs and constant forces on the right, things related to the dynamic variable and its derivatives on the left). You must also write out the mass moment of inertia of the system about the point \( Q \), \( I_Q \), in terms of the given variables. **C**: Convert the equation of motion so that it is in terms of \( x_1 \), the linear coordinate associated with the center of the disk. Again, be sure it is in input-output standard form.
Transcribed Image Text:**For this problem,** refer to Figure 2 below. A disk with uniformly distributed mass \( m \), radius \( R \), and center of mass at point \( O \) is connected to a combination of springs at point \( P \), which are then connected to a fixed wall. The disk rolls without slipping at point \( Q \) along an inclined plane that is at an angle \( \alpha \) from the horizontal. Gravity acts in the vertical direction (towards the bottom of the page). \( x_1 \) is the linear coordinate of the point \( O \) along the inclined plane. The positive direction of \( x_1 \) is as shown. When the springs are undeflected, \( x_1 = 0 \). An angle \( \theta \), about the instant center of rotation, is shown. You may assume that the motion (and therefore angle \( \theta \)) is small. **Figure Description:** - A disk is shown rolling along an inclined plane with an angle \( \alpha \). - The disk has a center of mass at point \( O \) and is connected to springs at point \( P \). - The springs are connected to each other and then to a fixed wall, labeled as a “Massless structure between springs.” - An angle \( \theta \) is depicted between a line perpendicular to the incline and a line connecting the point of contact and the center of the disk. - Other key points and dimensions include the radius \( R \), and the linear coordinate \( x_1 \). **Your Tasks:** **A**: Draw the Free Body Diagram (FBD) for the disk. Don’t forget the forces at point \( Q \). **B**: Derive the equation of motion with \( \theta \) as the dynamic variable. Be sure to put it in input-output standard form (inputs and constant forces on the right, things related to the dynamic variable and its derivatives on the left). You must also write out the mass moment of inertia of the system about the point \( Q \), \( I_Q \), in terms of the given variables. **C**: Convert the equation of motion so that it is in terms of \( x_1 \), the linear coordinate associated with the center of the disk. Again, be sure it is in input-output standard form.
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