The question asks which factor would NOT increase the acceleration of a ball going down an inclined plane.
The correct answer is the mass of the ball. However, I didn't even read that answer because the first answer choice was the angle of the incline.
I thought that increasing the angle of incline would decrease the acceleration and decreasing the angle would increase the acceleration.
Can you please explain?
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Be careful with questions like this. Our intuition, as humans, is not particularly good when it comes to changing angles and thinking about the effects. Since it mentions acceleration in the question stem, think... force!
Three steps:
1. Make a free-body diagram.
2. Write Newton's Second Law.
3. Plug and chug (or, in this case, think about the variables they mention in the answers).
For step 1, let's think about the forces we have:
-Gravity. It will point straight down.
-The normal force, pointing 90 degrees from the incline.
-Friction, if it exists. It will point up the plane, if the object is going down the plane.
Since we don't have the forces acting in two directions only (like x/y axes), we'll have to break the gravity up into components. The force of gravity acting parallel to the plane (down the plane) will be equal to mg sinθ; the force of gravity acting perpendicular to the plane (opposite the normal force) will be equal to mg cosθ. If you don't remember why these components are what they are, make sure to check out the section on inclined planes in the Review Notes or Lesson Book.
So, let's move onto Newton's Second Law. The dimension we're concerned with here is everything parallel to the plane. This would be F = ma, where the net force is equal to mg sinθ - f (which is the force of gravity parallel to the plane, minus the friction. So mg sinθ - f = ma.
Remember that friction can also be expressed as the coefficient of friction, μ, times the normal force. We also pointed out that the normal force is equal and opposite to the component of gravity pointing perpendicular to the plane (mg cosθ). So, friction = μN = μmg cosθ.
Putting that into our equation, we have mg sinθ - μmg cosθ = ma. We can cancel out m, and have just g sinθ - μg cosθ = a. This is perfect for the question -- let's see how each variable will affect acceleration.
Starting with A (increasing the angle of inclination), what will that affect? It will affect θ throughout. As θ increases from 0 to 90 degrees, sinθ also increases (from 0 to 1); conversely, cosθ decreases (from 1 to 0). So acceleration must be increasing -- gsinθ is increasing, and we're subtracting a smaller and smaller number with μg cosθ. So increasing the angle increases the acceleration. Also, think intuitively -- the acceleration when θ=0 should be zero (it's flat on a table), and when θ=90, acceleration will be equal to g, since the ball will be in freefall then.
Now for B and C. Decreasing the friction will decrease μ, so that will increase acceleration (we'll be subtracting a smaller number). We didn't talk about air resistance, but removing that drag force should also logically increase the acceleration.
Finally, for D, note that mass actually cancelled out of our equation. Whether we increase the mass or decrease it, the acceleration shouldn't change. In reality, this scenario is very common. Mass often cancels out in kinematics equations, since most forces are dependent on the mass (gravity, for example) and we use the equation F = ma. We can often divide both sides by m once we've gotten it set up.
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