Minimum Applied Force for Block and Cart Dynamics Problem

[Felafel]

Homework Statement

I've solved it already, I think. I'm just not sure about the result.

There is a block (B), which is touching a cart (C) on one side.
Let an external force, parallel to the surface, $\vec{F_a}$ be applied on B

mass of B = m; mass of C = M; static friction coefficient between B and C = μ.

Taking no notice of the ground's friction, what is the minimum value of $\vec{F_a}$ such that the block doesn't fall?

The Attempt at a Solution

After drawing the free-body diagram for B, i see:
$\vec{F_s}$ (static friction force) $\leq m \cdot \vec{g}$
and being $\vec{F_s}=μ \cdot \vec{F_N}$ i get $\vec{F_N}= \frac{m \cdot \vec{g}}{μ}$
$\vec{F_a}=\vec{F_N} + \vec{F_f}$ the latter being the force applied to C, which makes it move.
$\vec{F_f}=\frac{\vec{F_N}}{M} * m$ . So,
$\vec{F_a}=\frac{m \cdot g}{μ}+ \frac{m^2 \cdot g}{μ \cdot M}$

is it okay?

 

[ehild]

You made some little errors.

Homework Statement

I've solved it already, I think. I'm just not sure about the result.

There is a block (B), which is touching a cart (C) on one side.
Let an external force, parallel to the surface, $\vec{F_a}$ be applied on B

mass of B = m; mass of C = M; static friction coefficient between B and C = μ.

Taking no notice of the ground's friction, what is the minimum value of $\vec{F_a}$ such that the block doesn't fall?

The Attempt at a Solution

After drawing the free-body diagram for B, i see:
$\vec{F_s}$ (static friction force) $\leq m \cdot \vec{g}$

$\vec{F_s}$ (static friction force) $=-m \cdot \vec{g}$

and being $\vec{F_s}=μ \cdot \vec{F_N}$ i get $\vec{F_N}= \frac{m \cdot \vec{g}}{μ}$

${F_s}\leq \mu \cdot {F_N}$

$\vec{F_a}=\vec{F_N} + \vec{F_f}$ the latter being the force applied to C, which makes it move.
$\vec{F_f}=\frac{\vec{F_N}}{M} * m$ . So,
$\vec{F_a}=\frac{m \cdot g}{μ}+ \frac{m^2 \cdot g}{μ \cdot M}$

is it okay?

The minimum value of F_a is the question. So $F_a\geq\frac{m \cdot g}{μ}+ \frac{m^2 \cdot g}{μ \cdot M}$

ehild

 

[vela]

$\vec{F_a}=\vec{F_N} + \vec{F_f}$ the latter being the force applied to C, which makes it move.
$\vec{F_f}=\frac{\vec{F_N}}{M} * m$.

Can you explain these two steps? I don't follow what you did here.

 

[haruspex]

$\vec{F_a}=\vec{F_N} + \vec{F_f}$ the latter being the force applied to C, which makes it move.

As vela notes, this is wrong. Try introducing an unknown for the acceleration of the system and developing the F=ma equation for each body separately.

 

[ehild]

{μ}##
$\vec{F_a}=\vec{F_N} + \vec{F_f}$ the latter being the force applied to C, which makes it move.
$\vec{F_f}=\frac{\vec{F_N}}{M} * m$ . So,
$\vec{F_a}=\frac{m \cdot g}{μ}+ \frac{m^2 \cdot g}{μ \cdot M}$

is it okay?

You meant by F_f the resultant force acting on B instead of C, didn't you?

haruspex: The OP solved the problem, but made some little errors when typing in. The result for the minimum applied force is correct, except the vector sign.

ehild