Calculating Outward Normal Vector on Frictionless Ramp Inclined θ Degrees

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In summary, the conversation discusses finding the outward normal vector to the surface of an inclined ramp and using it to calculate the force in the direction of the incline. The normal vector is found to have coordinates of x = |g|cosθsinθ and y = |g|cos2θ, with a missing mass in the equations. It is also mentioned that using coordinates parallel and perpendicular to the incline surface may be more useful.
  • #1
amcavoy
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I was trying to figure out how to get the outward normal vector to the surface of a ramp inclined θ degrees from the horizontal. Say that a block of mass "m" is on the surface and the surface is frictionless. When I draw the free-body diagram, I come up with a downward force of <0,-mg>. To calculate the force in the direction of the incline, I first want to find the normal vector to add to <0,-mg>. Is this the correct way to do this? It looks like |n|=|g|cosθ, but I cannot find the coordinates of this vector. Any ideas?

Edit: Working it out, I came up with the following for the coordinates of the outward normal vector:

x = |g|cosθsinθ
y = |g|cos2θ

Are these correct?
 
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  • #2
You got it--except for a missing mass (you left out the "m" in your equations). (Note that it's often more useful to use coordinates parallel and perpendicular to the incline surface, rather than vertical and horizontal components.)
 
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  • #3
Oh yeah, the m! Anyways, thanks for the response :smile:.
 

1. What is an outward normal vector?

An outward normal vector is a vector that is perpendicular to a surface and points away from the surface, indicating the direction in which forces act on the surface. In the context of a frictionless ramp inclined θ degrees, the outward normal vector would be perpendicular to the surface of the ramp and would point away from the ramp.

2. How do you calculate the outward normal vector on a frictionless ramp inclined θ degrees?

To calculate the outward normal vector on a frictionless ramp inclined θ degrees, you can use the formula: N = mgcosθ, where N is the magnitude of the outward normal vector, m is the mass of the object on the ramp, g is the acceleration due to gravity, and θ is the angle of inclination of the ramp.

3. Why is the outward normal vector important in the context of a frictionless ramp inclined θ degrees?

The outward normal vector is important because it helps us understand the direction of forces acting on an object on the ramp. In the case of a frictionless ramp, the only force acting on the object is the force due to gravity, and the outward normal vector helps us determine the magnitude and direction of this force.

4. How does the value of θ affect the calculation of the outward normal vector?

The value of θ directly affects the calculation of the outward normal vector. As θ increases, the magnitude of the outward normal vector also increases, meaning that the force due to gravity acting on the object also increases. Conversely, as θ decreases, the magnitude of the outward normal vector and the force due to gravity decrease as well.

5. Can the outward normal vector ever be negative?

No, the outward normal vector cannot be negative. It represents a direction, not a value, and is always perpendicular to the surface of the ramp and points away from it. This means that it can only have positive values in the direction away from the surface. In mathematical terms, it is a magnitude and direction vector, not a scalar value.

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