The minimum height of the incline required for the disc to loop the groove is B.9R/4 . Therefore , B.9R/4 is correct .
Let's analyze the scenario and determine the minimum height required for the disc to loop the groove.
Initial Energy: At the top of the incline, the disc has potential energy due to its height, mgH.
Energy Conversion: As it rolls down the incline, the potential energy gets converted into kinetic energy and rotational energy.
Minimum Height: To complete the loop, the disc needs enough kinetic energy at the bottom to overcome the change in potential energy required to reach the top of the groove.
Here's how we can find the minimum height:
Step 1: Apply Energy Conservation
At the bottom of the incline, just before entering the groove:
Potential Energy + Kinetic Energy + Rotational Energy = Potential Energy at the top of the groove.
Step 2: Define the variables
H: Height of the incline
R: Radius of the groove
m: Mass of the disc
v: Velocity of the disc at the bottom of the incline
I: Moment of inertia of the disc
Step 3: Express the energies in terms of the variables
Potential Energy at the top = mgh
Potential Energy at the top of the groove = mg(2R)
Kinetic Energy = 1/2 mv^2
Rotational Energy = 1/2 Iω^2
Moment of inertia of a solid disc = 1/2 mr^2
Rolling without slipping means v = ωr
Step 4: Substitute the expressions in the energy conservation equation
mgh + 1/2 mv^2 + 1/2(1/2 mr^2)(v/r)^2 = mg(2R)
Step 5: Simplify and solve for H
Cancel common factors and rearrange the equation:
H = 2R + (v^2)/2g
Since the disc needs enough energy to loop the groove, the velocity at the bottom should be enough for it to reach the top of the groove:
v^2 = 2gR
Substitute this back into the equation for H:
H = 2R + 2R = 4R
Therefore, the minimum height of the incline required for the disc to loop the groove is 9R/4.
Question
A disc is at rest at the top of a rough inclined plane. It rolls without slipping. At the bottom of inclined plane there is a smooth vertical groove of radius ‘R’. In order to loop the groove. the minimum height of incline required is
A. 15R/4
B. 9R/4
C. 5R/2
D. 7R/5