Can someone please show me how to derive this...I am going crazy I have tried everyhthing that I can think of with this equation..even the Range Equation and that doesnt work! So please! I am dying and aahaha! How does this work in terms of mass and L? I think L has to be radius though, and you use something to do with centripetal force? HELP
a= gtan?
Forget about the rubber stopper spinning and that it has a radius. Just treat it as a mass at the end of a string. The string exerts a tension force T on the stopper. When the car accelerates, the string deviates from a vertical line by an agle theta. Write down the force balance on the stopper.
T cos theta = m a
T sin theta = m g
Divide the first equation by the second:
(sin theta)/(cos theta) = tan theta
= a/g
Note that the mass cancels out.
I get it, but I think it needs to be T cos theta=mg and T sin theta =ma and then when you divide you divide Tsintheta=ma/T cos theta=mg
which leaves u with Sintetha=a/Cos theta=g
so Tantheta=a/g and then rearrange to get a=gtantheta... so thanks! a lot! This makes sense now!
that should be a= gtan? (theta, not ?)
What is the physical situation that this formula is supposed to apply to? What is theta? One can't derive a formula of this type without knowing the problem
Thats my thoughts exaclty! It is this rubber stopper of mass m is in a car and then when the car accelerates it makes an angle with the veritcal which is theta. We are supposed to find a=gtantheta in terms of mass and L, which must be radius, cause the rubber stopper (which is on a string on a rear view mirror) is spinning, so something to do with centripetal motion...this is all I can gather...please try and help me with this, I know its a lot to ask, but Im drowning with this, its the only question I cant even begin to guess!
You are right; I got my sines and cosines mixed up, and then I mixed up the equaitons to cancel out my mistake. I was in too big a hurry. Nice work catching my error(s)!
Sure, I can help you derive the equation using the given information. The equation you mentioned, a = gtanθ, relates the acceleration (a) of an object to the gravitational acceleration (g) and the angle of inclination (θ). It seems like you're trying to relate this equation to mass (m) and L (which you think represents the radius).
To begin, let's clarify a few things:
- The equation a = gtanθ involves the plane's angle of inclination (θ), not the radius (L).
- The acceleration (a) in this equation refers to the component of acceleration parallel to the incline, not the total acceleration.
- The mass (m) of the object is missing from the given equation.
To derive a correct equation that includes mass and radius, we need to use a different approach. We can use the concepts of centripetal force and circular motion.
Consider an object of mass (m) moving in a horizontal circle of radius (r) with a constant speed (v). The object experiences centripetal force (Fc) that keeps it moving in a circular path.
The centripetal force can be defined as the mass times the centripetal acceleration:
Fc = m * ac
The centripetal acceleration (ac) can be calculated using the equation:
ac = v^2 / r
Substituting the expression for centripetal acceleration into the centripetal force equation:
Fc = m * (v^2 / r)
Since we are given the gravitational acceleration (g) and assuming the object is at rest on an inclined plane, it follows that the vertical component of the weight of the object is equal to the force required to balance it:
mg * sinθ = Fc
Substituting the centripetal force in terms of mass, velocity, and radius:
mg * sinθ = m * (v^2 / r)
Simplifying by canceling out the mass:
g * sinθ = v^2 / r
Rearranging the equation to solve for v^2:
v^2 = g * r * sinθ
Now, we have an equation that relates the speed of the object (v) to the gravitational acceleration (g), radius (r), and the angle of inclination (θ). Note that this equation doesn't involve the tangent function.
If you need to relate the acceleration (a) to the gravitational acceleration (g), you can use the relationship between acceleration and speed:
a = v^2 / r
Substituting the expression for v^2 from the previous equation:
a = g * r * sinθ / r
Simplifying:
a = g * sinθ
So, the correct equation relating the acceleration (a) to the gravitational acceleration (g) and the angle of inclination (θ) is indeed a = g * sinθ.
I hope this explanation helps you understand the correct derivation and the concepts involved. Let me know if you have any further questions!