A small block is released from rest at the top of a long frictionless ramp that is inclined at an angle of 36.9° above the horizontal. You measure that a small block travels a distance 12.0 m down the incline in 8.20 s. What is the value of g, the acceleration due to gravity on this planet?

To find the value of g, the acceleration due to gravity on the planet, we can use the equations of motion for motion along an inclined plane.

We have the following information:
- Distance traveled down the incline, d = 12.0 m
- Time taken, t = 8.20 s
- Angle of inclination, θ = 36.9°

The equation to calculate the distance traveled down the incline is given by:
d = (1/2) * g * t^2 * sin(2θ)

Rearranging the equation, we can solve for g:
g = (2 * d) / (t^2 * sin(2θ))

Now, we can substitute the given values into the equation and calculate the value of g.

g = (2 * 12.0) / (8.20^2 * sin(2 * 36.9))

g = (24.0) / (8.20^2 * sin(73.8))

Calculating sin(73.8) using a scientific calculator, we get:

sin(73.8) ≈ 0.9511

Substituting the values:

g = (24.0) / (8.20^2 * 0.9511)

Calculating 8.20^2 using a calculator, we get:

g = (24.0) / (67.24 * 0.9511)

Calculating 67.24 * 0.9511 using a calculator:

g ≈ (24.0) / (63.9488)

Finally, calculating 24.0 / 63.9488:

g ≈ 0.375 m/s^2

Therefore, the value of g, the acceleration due to gravity on this planet, is approximately 0.375 m/s^2.