Question

scenario
two blocks are being pushed across a surface with an external force f, as shown in the figure at the right. the mass m₂ of block 2 is greater than the mass m₁ of block 1. the blocks begin at rest. the surface is smooth enough that the frictional forces between the surface and the block can be neglected.
using representations
part a:
the dots below represent the two blocks. draw free - body diagrams showing and labelling the forces (not components) exerted on each block. draw the relative lengths of all vectors to reflect the relative magnitudes of all the forces. each force must be represented by a distinct arrow starting on and pointing away from the dot.
quantitative analysis
part b:
derive the magnitude of the acceleration of block 2. express your answers in terms of m₁, m₂, g, and f.
σfₓ = maₓ
the sum of the external forces on the system will be equal to the mass of the system times the acceleration of the system.
the net external force (in the horizontal direction) is f_push.
the mass of the system is the sum of the two masses.
the acceleration of the system is then:
and since mass 2 will have the same acceleration as the system, the acceleration of mass 2 is:

Explanation:

Step1: Identify the net - force on the system

The net external force acting on the two - block system in the horizontal direction is the applied force $F$ since the surface is smooth and there is no friction. The vertical forces (weight and normal force) cancel each other out. So, $\sum F_x=F$.

Step2: Determine the mass of the system

The mass of the two - block system is the sum of the masses of the two blocks, $m = m_1 + m_2$.

Step3: Apply Newton's second law

According to Newton's second law $\sum F_x=ma_x$. Substituting $\sum F_x = F$ and $m=m_1 + m_2$ into the equation, we get $F=(m_1 + m_2)a$.

Step4: Solve for the acceleration

We can solve the equation $F=(m_1 + m_2)a$ for the acceleration $a$. Rearranging the equation gives $a=\frac{F}{m_1 + m_2}$. Since the two blocks move together as a single unit (no relative motion between them), the acceleration of block 2 is the same as the acceleration of the system.

Answer:

$a=\frac{F}{m_1 + m_2}$