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A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,?
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A ring of mass m and radius R rolls (without slipping) down an incline...
Introduction
When a ring rolls down an inclined plane without slipping, it converts its gravitational potential energy into kinetic energy, which includes both translational and rotational components.
Energy Conservation
- The initial potential energy (PE) at height h is given by:
- PE_initial = mgh
- When the ring reaches the base, this potential energy is converted into kinetic energy (KE):
- KE_total = KE_translational + KE_rotational
Kinetic Energy Components
- The translational kinetic energy (KE_translational) is:
- KE_translational = (1/2)mv^2
- The rotational kinetic energy (KE_rotational) is:
- KE_rotational = (1/2)Iω^2
- For a ring, the moment of inertia (I) is given by I = mR^2.
Relationship Between Linear and Angular Velocity
- The ring rolls without slipping, so:
- v = Rω
Substituting for Angular Velocity
- Using the relationship between linear and angular velocity, we can express ω:
- ω = v/R
Energy Equation
- Setting initial potential energy equal to total kinetic energy at the base:
- mgh = (1/2)mv^2 + (1/2)(mR^2)(v/R)^2
- Simplifying this gives:
- mgh = (1/2)mv^2 + (1/2)mv^2 = mv^2
Finding Linear Velocity
- Cancelling m and rearranging:
- gh = v^2, hence:
- v = sqrt(gh)
Calculating Angular Velocity
- Substituting back to find ω:
- ω = v/R = sqrt(gh)/R
Conclusion
The angular velocity when the ring reaches the base of the incline is given by:
ω = sqrt(gh)/R.
When a ring rolls down an inclined plane without slipping, it converts its gravitational potential energy into kinetic energy, which includes both translational and rotational components.
Energy Conservation
- The initial potential energy (PE) at height h is given by:
- PE_initial = mgh
- When the ring reaches the base, this potential energy is converted into kinetic energy (KE):
- KE_total = KE_translational + KE_rotational
Kinetic Energy Components
- The translational kinetic energy (KE_translational) is:
- KE_translational = (1/2)mv^2
- The rotational kinetic energy (KE_rotational) is:
- KE_rotational = (1/2)Iω^2
- For a ring, the moment of inertia (I) is given by I = mR^2.
Relationship Between Linear and Angular Velocity
- The ring rolls without slipping, so:
- v = Rω
Substituting for Angular Velocity
- Using the relationship between linear and angular velocity, we can express ω:
- ω = v/R
Energy Equation
- Setting initial potential energy equal to total kinetic energy at the base:
- mgh = (1/2)mv^2 + (1/2)(mR^2)(v/R)^2
- Simplifying this gives:
- mgh = (1/2)mv^2 + (1/2)mv^2 = mv^2
Finding Linear Velocity
- Cancelling m and rearranging:
- gh = v^2, hence:
- v = sqrt(gh)
Calculating Angular Velocity
- Substituting back to find ω:
- ω = v/R = sqrt(gh)/R
Conclusion
The angular velocity when the ring reaches the base of the incline is given by:
ω = sqrt(gh)/R.
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A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,?
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A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,?.
A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is,?.
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