Question

A uniform sphere of mass m is positioned between two horizontal planks A and B of mass 2 m and m respectively as shown. Ground surface is smooth and friction between planks and sphere is sufficient so that no slipping takes place. Horizontal forces $F_1$ and $F_2$ are applied on plank so that acceleration of sphere is zero. Plank B is moving towards right with acceleration $a$. Which of the following is/are correct?

• A. $\text{Both planks have same acceleration}$
• B. $\text{Net force on sphere is non-zero}$
• C. $F_1+F_2=\frac{17}{5}ma$
• D. $\text{friction}=5ma$

Answer 1

The correct option is C $F_1+F_2=\frac{17}{5}ma$

Since, sphere is at rest, both friction at top and bottom will be equal as acceleration of sphere is zero. Acceleration of both planks -
$a_1=a_2=\alpha R=a\therefore$ No slipping
$F_1-f=ma$ ...(1)
$F_2-f=2ma$ ...(2)
$2fR=\frac{2}{5}m{R}^2\alpha$
$\Rightarrow 2f=\frac{2}{5}ma$ ...(3)
Adding equations (1), (2) and (3),
$F_1-f+F_2-f+2f=ma+2ma+\frac{2}{5}ma$
$F_1+F_2=(3+\frac{2}{5})ma=\frac{17}{5}ma$

Why this question? Tip: The condition for no slipping can be applied at all the common interfaces- watch out- you are never short of equations.