Question

An annular disc of mass M, inner radius a and outer radius b is placed on a horizontal surface with a coefficient of friction μ, as shown in the figure. At some time, an impulse J₀ \(\hat{x}\) is applied at a height of h at the center of the disc. If h=h_m, then the disc rolls without slipping along the x-axis, which of the following statement is/are correct? 

• A. For h=h_m the initial angular velocity doesn't depend on the inner radius a.
• B. For μ≠0 & a\(\rightarrow\)b, h_m=b
• C. For μ=0 & h=0 the wheel always slides without rolling
• D. For μ≠0 & a\(\rightarrow\)0, h_m=\(\frac{b}{2}\)

Answer 1

The Correct Option is A, B, C, D

Angular impulse about \(CM=\Delta L\)
\(J_0\,h=I_{cm}\omega\)
\(\omega=\frac{J_0h}{I_{cm}}\)
Linear impulse = \(\Delta P\)
\(J_0=\Delta P\)
\(J_0=MV-0\)
\(V=\frac{J_0}{M}\)
As disc is pure rolling,
\(V=\omega b\)
\(\omega=\frac{V}{b}\)
\(\omega=\frac{J_0}{Mb}\)
\(\omega\) is independent of radius a…… (Ans.1)
Now, If \(a\rightarrow b\),  then the annular disc is a ring of radius b
\(I_{cm}=Mb^2\)
\(\omega=\frac{J_0h}{Mb^2}\)
\(\frac{J_0}{Mb}=\frac{J_0h}{Mb^2}\)
\(h=b\)
\(\therefore h_m=b\)…….(Ans.2)

if h=0 and \(\mu\)=0,
as h=0 \(\,\,\,\,\omega=\frac{J_0h}{Mb^2}\) = 0
but \(V=\frac{J_0}{M}\)………(Ans.3)
If a\(\rightarrow\)0then annular disc is disc of radius b
\(I_{cm}=\frac{Mb^2}{2}\)
\(\omega=\frac{J_0h\times2}{Mb^2};h=\frac{b}{2};h_m=\frac{b}{2}\)……..(Ans.4)