Question
A hollow sphere of mass $$2\,kg$$ is kept on a rough horizontal surface. A force of $$10\,N$$ is applied at the centre of the sphere as shown in the figure. Find the minimum value of $$\mu $$ so that the sphere starts pure rolling. (Take $$g = 10\,m/{s^2}$$ )
A hollow sphere of mass $$2\,kg$$ is kept on a rough horizontal surface. A force of $$10\,N$$ is applied at the centre of the sphere as shown in the figure. Find the minimum value of $$\mu $$ so that the sphere starts pure rolling. (Take $$g = 10\,m/{s^2}$$ )
A.
$$\sqrt 3 \times 0.16$$
B.
$$\sqrt 3 \times 0.08$$
C.
$$\sqrt 3 \times 0.1$$
D.
Data insufficient
Answer :
$$\sqrt 3 \times 0.08$$
Solution :
$$\eqalign{ & 10\cos {30^ \circ } - f = 2a\,......\left( {\text{i}} \right) \cr & \tau = I\alpha \cr & \Rightarrow fr = \frac{2}{3} \times 2 \times {r^2} \times \alpha \,......\left( {{\text{ii}}} \right)\,\,\left( {{\text{where }}r{\text{ is radius of sphere}}} \right) \cr} $$
From (i) and (ii), we get
$$\eqalign{ & f = 2\sqrt 3 \,{\text{newton,}} \cr & N = 20 + 10\sin {30^ \circ } = 25 \cr & f = \mu N \Rightarrow \mu = \frac{f}{N} = \frac{{2\sqrt 3 }}{{25}} = 0.08 \times \sqrt 3 \cr} $$
$$\eqalign{ & 10\cos {30^ \circ } - f = 2a\,......\left( {\text{i}} \right) \cr & \tau = I\alpha \cr & \Rightarrow fr = \frac{2}{3} \times 2 \times {r^2} \times \alpha \,......\left( {{\text{ii}}} \right)\,\,\left( {{\text{where }}r{\text{ is radius of sphere}}} \right) \cr} $$
From (i) and (ii), we get
$$\eqalign{ & f = 2\sqrt 3 \,{\text{newton,}} \cr & N = 20 + 10\sin {30^ \circ } = 25 \cr & f = \mu N \Rightarrow \mu = \frac{f}{N} = \frac{{2\sqrt 3 }}{{25}} = 0.08 \times \sqrt 3 \cr} $$
