Ranking applied forces

Problem

Three blocks are pushed on a smooth horizontal plane:

• A: $m = 2\;\text{kg}$ with acceleration $3\;\text{m/s}^2$;
• B: $m = 3\;\text{kg}$ with acceleration $2\;\text{m/s}^2$;
• C: $m = 4\;\text{kg}$ with acceleration $2\;\text{m/s}^2$.

Rank the applied force from smallest to largest.

Solution

Since the plane is smooth, the only horizontal force on each block is the applied one, and Newton’s second law $F = m\,a$ holds. Let us calculate the three values:

$F_A = m_A\,a_A = 2 \cdot 3 = 6\;\text{N}$

$F_B = m_B\,a_B = 3 \cdot 2 = 6\;\text{N}$

$F_C = m_C\,a_C = 4 \cdot 2 = 8\;\text{N}$

The ranking reasoning shows two useful lessons: A and B, despite having different mass and acceleration, require the same force because the product $m\,a$ coincides; C and B have the same acceleration, but C has greater mass and therefore requires more force. Ranking from smallest to largest:

$\ev{F_A = F_B < F_C}$

Links

Topics: Dynamics Concepts: Newton’s second law Skills: Ranking reasoning