Constraint Equation of a rolling of cylinder

Click For Summary
SUMMARY

The constraint equation for a cylinder of radius 'r' rolling inside another cylinder of radius 'R' without slipping is correctly defined as rθ = (R-r)φ. In this equation, θ represents the angle of rotation of the inner cylinder, while φ denotes the angle subtended at the center of the outer cylinder. The confusion arises from the assumption that rθ should equal Rφ; however, the correct relationship accounts for the difference in radii, leading to the equation involving (R-r). For example, with R = 4 and r = 1, the smaller cylinder completes one full rotation after rolling around the inner circumference of the larger cylinder.

PREREQUISITES
  • Understanding of rotational motion and angular displacement
  • Familiarity with the concept of rolling without slipping
  • Basic knowledge of geometry related to circles and cylinders
  • Ability to manipulate and solve equations involving angles and radii
NEXT STEPS
  • Explore the principles of rolling motion in physics
  • Study the relationship between linear and angular velocity
  • Learn about the geometry of cylinders and their properties
  • Investigate applications of constraint equations in mechanical systems
USEFUL FOR

Students of physics, mechanical engineers, and anyone interested in understanding the dynamics of rolling motion and constraint equations in cylindrical systems.

Molar
Messages
45
Reaction score
19
I was going through this example from by book, where a cylinder of radius 'r' is rolling inside of another cylinder of radius 'R' without slipping.It says the constraint equation should be
rθ = (R-r)φ
where θ = angle of rotation of cylinder of radius 'r'
φ = angle subtended at the centre of the cylinder of radius 'R'

I am not getting this. Shouldn't it be rθ = Rφ ...? To me it looks like if we deduct 'r' from 'R' then
rθ will be greater than (R-r)φ . Please correct me.
Thanks.
 
Physics news on Phys.org
Molar said:
I was going through this example from by book, where a cylinder of radius 'r' is rolling inside of another cylinder of radius 'R' without slipping.It says the constraint equation should be
rθ = (R-r)φ
where θ = angle of rotation of cylinder of radius 'r'
φ = angle subtended at the centre of the cylinder of radius 'R'

I am not getting this. Shouldn't it be rθ = Rφ ...? To me it looks like if we deduct 'r' from 'R' then
rθ will be greater than (R-r)φ . Please correct me.
Thanks.

Try an example where ##R = 4## and ##r = 1##

How many times does the smaller cylinder rotate to circumscribe the inside of the larger cylinder once?
 

Similar threads

Constraints on a hoop rolling on a cylinder
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Moment of inertia of human body with limbs at an angle
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Why is T+f=Ma in Cylinder Roll Without Gliding?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Lagrangian mechanics: sphere inside a cylinder
  • · Replies 6 ·
Replies
6
Views
11K
Undergrad Solving Wheel Coming to a Stop: FBD, Friction & Hysteresis
  • · Replies 22 ·
Replies
22
Views
5K
Undergrad Some questions about toppling of a set of two cylinders
  • · Replies 10 ·
Replies
10
Views
7K
A cylinder of snow rolls down a hill gathering more snow -- calculate its speed
  • · Replies 39 ·
2
Replies
39
Views
3K
Undergrad Is Mechanical Energy Conservation Free of Ambiguity - follow up
  • · Replies 77 ·
3
Replies
77
Views
5K
Motion of Rolling Cylinder in Fixed Cylinder: Confusing Constraint Condition
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Constraints in Rotation Matrix
  • · Replies 2 ·
Replies
2
Views
2K