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A bead of mass m can slide without friction a... Join the Discussion on the App A bead of mass m can slide without friction along a massless rod kept at 45° with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed ω. At any instant, r is the distance of the bead from the origin. The momentum conjugate to r is?
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Understanding the System
The scenario involves a bead sliding on a massless rod inclined at 45° to the vertical. The rod is rotating with a constant angular speed ω about the vertical axis. The bead's motion can be analyzed using concepts from classical mechanics.
Key Variables
- m: Mass of the bead
- r: Distance of the bead from the origin
- ω: Angular speed of the rod
Angular Momentum and Conjugate Momentum
In this context, the momentum conjugate to the radial coordinate r can be understood in terms of angular momentum. The bead experiences a centripetal force due to its circular motion around the axis of rotation.
Expression for Momentum Conjugate to r
The momentum conjugate to r, denoted as p_r, can be derived from the kinetic energy of the bead in the rotating frame. The kinetic energy (T) of the bead can be expressed as:
- T = (1/2) m (v^2)
Where v is the tangential speed of the bead given by v = ω * r.
Thus, substituting for v:
- T = (1/2) m (ω * r)^2
The conjugate momentum p_r is defined as the derivative of the kinetic energy with respect to the velocity associated with the radial coordinate. In this case:
- p_r = m * v = m * (ω * r)
Conclusion
Therefore, the momentum conjugate to r is given by the expression:
- p_r = m * ω * r
This expression captures the relationship between the radial position of the bead and its momentum due to the rotation of the system.
The scenario involves a bead sliding on a massless rod inclined at 45° to the vertical. The rod is rotating with a constant angular speed ω about the vertical axis. The bead's motion can be analyzed using concepts from classical mechanics.
Key Variables
- m: Mass of the bead
- r: Distance of the bead from the origin
- ω: Angular speed of the rod
Angular Momentum and Conjugate Momentum
In this context, the momentum conjugate to the radial coordinate r can be understood in terms of angular momentum. The bead experiences a centripetal force due to its circular motion around the axis of rotation.
Expression for Momentum Conjugate to r
The momentum conjugate to r, denoted as p_r, can be derived from the kinetic energy of the bead in the rotating frame. The kinetic energy (T) of the bead can be expressed as:
- T = (1/2) m (v^2)
Where v is the tangential speed of the bead given by v = ω * r.
Thus, substituting for v:
- T = (1/2) m (ω * r)^2
The conjugate momentum p_r is defined as the derivative of the kinetic energy with respect to the velocity associated with the radial coordinate. In this case:
- p_r = m * v = m * (ω * r)
Conclusion
Therefore, the momentum conjugate to r is given by the expression:
- p_r = m * ω * r
This expression captures the relationship between the radial position of the bead and its momentum due to the rotation of the system.
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A bead of mass m can slide without friction a... Join the Discussion on the App A bead of mass m can slide without friction along a massless rod kept at 45° with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed ω. At any instant, r is the distance of the bead from the origin. The momentum conjugate to r is?
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A bead of mass m can slide without friction a... Join the Discussion on the App A bead of mass m can slide without friction along a massless rod kept at 45° with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed ω. At any instant, r is the distance of the bead from the origin. The momentum conjugate to r is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A bead of mass m can slide without friction a... Join the Discussion on the App A bead of mass m can slide without friction along a massless rod kept at 45° with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed ω. At any instant, r is the distance of the bead from the origin. The momentum conjugate to r is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A bead of mass m can slide without friction a... Join the Discussion on the App A bead of mass m can slide without friction along a massless rod kept at 45° with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed ω. At any instant, r is the distance of the bead from the origin. The momentum conjugate to r is?.
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