Physics: slope factors vs angle, part1

Tutoring physics, you consider gravity against a slope. The tutor makes a comparison.

The following is by my understanding.

Let’s imagine one has to work on a slope of 22° and another of 35°. How much more likely is it to slip down the steeper one, and why?

The acceleration due to gravity is given by gsinθ, where θ is the angle of elevation (22° or 35° in this case). The grip of friction, holding the person in place on the slope, is calculated from gcosθ.

Importantly, to compare the 22° slope with the 35° one, the difference between sinθ on both slopes, then between cosθ on both, is key. To compare them, we will round to two decimal places.

sin22°=0.37, while sin35°=0.57. Therefore, the acceleration down the slope increases by about 54 percent from 22° to 35°.

cos22°=0.93, while cos35°=0.82. Therefore, friction decreases between 22° and 35°, but only by about 12 percent.

Therefore, it seems that what makes the 35° slope so much more challenging to stay in place on, compared to the 22° one, is the increase in acceleration down the slope moreso than the accompanying decrease in friction.

Source:

Giancoli,D.C. (1998). Physics, fifth edition. Prentice Hall.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.