Ranking smooth inclined planes

Problem

Four blocks slide on smooth inclined planes:

• (A) $10^\circ$;
• (B) $25^\circ$;
• (C) $45^\circ$;
• (D) $70^\circ$.

Rank the accelerations $a = g\sin\theta$ from smallest to largest.

Four smooth planes of increasing inclination from A to D.

Solution

Formula. On a smooth plane the only driving force is the component of the weight along the plane, so: $a = g\sin\theta$ The function $\sin\theta$ is increasing for $0^\circ < \theta < 90^\circ$: the steeper the plane, the greater the acceleration.

Case-by-case calculation (with $g = 9{,}81\ \mathrm{m/s}^2$). $a_A = g\sin 10^\circ \approx 9{,}81\cdot 0{,}174 \approx 1{,}70\ \mathrm{m/s}^2$ $a_B = g\sin 25^\circ \approx 9{,}81\cdot 0{,}423 \approx 4{,}15\ \mathrm{m/s}^2$ $a_C = g\sin 45^\circ \approx 9{,}81\cdot 0{,}707 \approx 6{,}94\ \mathrm{m/s}^2$ $a_D = g\sin 70^\circ \approx 9{,}81\cdot 0{,}940 \approx 9{,}22\ \mathrm{m/s}^2$

Increasing order. Since the angles increase and $\sin$ is monotonically increasing, the order follows directly from that of the angles: $\ev{A < B < C < D}$

Linked items

Topics: Dinamica Concepts: Forza peso · Seconda legge di Newton Skills: Ragionamento di ranking · Scomposizione sul piano inclinato Objects: Piano inclinato