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Is the minimum force necessary to pull a mass up the ramp same as the minimum force necessary to prevent the mass from sliding down the plane?
Mass on ramp question
- Thread starter sunshine02
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Consider a mass placed on an inclined plane (a ramp.) What forces are acting on it? Gravity of course, and thus the normal force as well. However, the normal force acts perpendicular to the plane, while gravity acts directly downward (perpendicular to the ground.) Thus, the normal force is equal only to the vector component (of the gravitational force) that is perpendicular to the plane. It is then obvious that the reason gravity compels the object down the plane is that there it has a non-zero vector component left over which runs parallel to the plane. The frictional force acts opposite this "leftover" force. All of these can be seen below:
Let's also remember two things: 1) If the mass at rest, ∑F = 0 and 2) if we are pulling the mass up the ramp at a constant speed then here too ∑F = 0.
Note that f represents the force necessary to prevent the mass from sliding down the ramp. This is not the same as the force we would need to apply to keep the mass from sliding down the ramp; if the coefficient of friction is great enough between the mass and the plane, we may not need to apply any additional force at all, friction could keep it at rest alone. If that's the case, then f = f_static = mgsinø. If that's not the case, and we do need to apply some additional force, then f = f_static < mgsinø and the magnitude of the additional force we would need to apply would be | f_add| = |mgsinø - f_static|.
If we pull the mass in the up the ramp, friction is now acting in the opposite direction and is no longer static friction, but kinetic friction (which will always have a lesser magnitude than static), so the force needed to keep it moving must be an additional force we supply. Whereas with a resting mass we only needed the force directed up the ramp to equal mgsinø, we now need it to equal f_kinetic + mgsinø.
To summarize, take the example of a rubber block on a concrete inclined plane (i.e. sufficient coeff. of friction that mass can rest when placed):
Mass at rest: magnitude of external force up plane needed to maintain rest = 0 AND magnitude of absolute force up plane needed to maintain rest = |mgsinø|
Mass being pulled: magnitude of external force up plane needed to maintain motion = |mgsinø + f_kinetic| = magnitude of absolute force up plane needed to maintain motion > 0
For a case of a cardboard block placed on a smooth metal inclined plane (i.e. insufficient coeff. of friction to maintain rest when placed):
Mass at rest: magnitude of external force up plane needed to maintain rest = magnitude of absolute force up plane needed to maintain rest = |mgsinø| > 0
Mass being pulled: magnitude of external force up plane needed to maintain motion = |mgsinø + f_kinetic| = magnitude of absolute force up plane needed to maintain motion > 0
TL;DR - No.
Let's also remember two things: 1) If the mass at rest, ∑F = 0 and 2) if we are pulling the mass up the ramp at a constant speed then here too ∑F = 0.
Note that f represents the force necessary to prevent the mass from sliding down the ramp. This is not the same as the force we would need to apply to keep the mass from sliding down the ramp; if the coefficient of friction is great enough between the mass and the plane, we may not need to apply any additional force at all, friction could keep it at rest alone. If that's the case, then f = f_static = mgsinø. If that's not the case, and we do need to apply some additional force, then f = f_static < mgsinø and the magnitude of the additional force we would need to apply would be | f_add| = |mgsinø - f_static|.
If we pull the mass in the up the ramp, friction is now acting in the opposite direction and is no longer static friction, but kinetic friction (which will always have a lesser magnitude than static), so the force needed to keep it moving must be an additional force we supply. Whereas with a resting mass we only needed the force directed up the ramp to equal mgsinø, we now need it to equal f_kinetic + mgsinø.
To summarize, take the example of a rubber block on a concrete inclined plane (i.e. sufficient coeff. of friction that mass can rest when placed):
Mass at rest: magnitude of external force up plane needed to maintain rest = 0 AND magnitude of absolute force up plane needed to maintain rest = |mgsinø|
Mass being pulled: magnitude of external force up plane needed to maintain motion = |mgsinø + f_kinetic| = magnitude of absolute force up plane needed to maintain motion > 0
For a case of a cardboard block placed on a smooth metal inclined plane (i.e. insufficient coeff. of friction to maintain rest when placed):
Mass at rest: magnitude of external force up plane needed to maintain rest = magnitude of absolute force up plane needed to maintain rest = |mgsinø| > 0
Mass being pulled: magnitude of external force up plane needed to maintain motion = |mgsinø + f_kinetic| = magnitude of absolute force up plane needed to maintain motion > 0
TL;DR - No.
