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 determine the value of g, the acceleration due to gravity on this planet, we can use the equation of motion for an object moving down an inclined plane:
\[d = \frac{1}{2}gt^2\]
where:
d = distance traveled down the incline (12.0 m)
g = acceleration due to gravity
t = time taken (8.20 s)
First, we need to find the time it would take for the object to fall freely (without the ramp). We can use the vertical motion equation: \[d = \frac{1}{2}gt^2\], where d is the height of the ramp.
Since the ramp is inclined at an angle of 36.9°, we can find d using trigonometry:
\[d = \text{height of the ramp} = \text{distance down the incline} \times \sin(\text{angle of inclination})\]
\[d = 12.0 \times \sin(36.9°)\]
Next, we can calculate the time taken for the object to fall freely:
\[d = \frac{1}{2}gt^2\]
\[12.0 \sin(36.9°) = \frac{1}{2}g t^2\]
Now, we can solve for t:
\[t = \sqrt{\frac{2d}{g}}\]
Substituting in the known values, t = 8.20 s and d = 12.0 sin(36.9°), we can rearrange the equation to solve for g:
\[g = \frac{2d}{t^2}\]
Now, substitute the given values to calculate g.
To find the value of g, the acceleration due to gravity on this planet, we can use the formula for the acceleration of an object on an inclined plane.
The formula is given by:
a = g * sin(θ)
Where:
a = acceleration down the incline
g = acceleration due to gravity (unknown)
θ = angle of inclination (36.9° in this case)
We are given the distance traveled down the incline (12.0 m) and the time taken (8.20 s). We can use these values to find the acceleration.
The formula to find acceleration (a) is:
a = (2 * d) / t^2
Where:
a = acceleration
d = distance traveled down the incline (12.0 m)
t = time taken (8.20 s)
Plugging in the values:
a = (2 * 12.0) / (8.20)^2
Calculating:
a = 0.292 m/s^2
Now, we can equate this value of acceleration (a) to the formula for the acceleration on the inclined plane to find g:
0.292 = g * sin(36.9°)
Rearranging the equation to solve for g, we get:
g = 0.292 / sin(36.9°)
Calculating this value:
g = 0.292 / sin(36.9°)
g ≈ 0.478 m/s^2
Therefore, the value of g, the acceleration due to gravity on this planet, is approximately 0.478 m/s^2.