# Radius of Curvature of particle

• danago
In summary, the particle travels a curved path with a radius of curvature of 130.4 meters at time t=5.443.

#### danago

Gold Member
A particle is fired into the air with an initial velocity of 60m/s at an angle of 54 degrees from the ground. At time t=5.443, what is the radius of curvature of the path traveled by the particle?

I started by coming up with a vector equation for the path traveled by the particle, using the point of launch as the reference point.

$$\overrightarrow r = (60t\cos 54)\widehat{\underline i } + (60t\sin 54 - 0.5g^2 )\widehat{\underline j }$$

The way i thought to approach this problem was to consider a circle which closely approximates the curve at the given instant.

For an infinitesimal change in time dt, the velocity along the path is given by:

$$v = r\frac{{d\theta }}{{dt}}$$

I thought that perhaps if i could calculate the speed along the path v and the angular velocity, then i could use those to calculate the radius of curvature at that instant. I am able to calculate the speed easily, but not so sure about the angular velocity.

Am i on the right track, or should i be taking a different approach?

Dan.

Looks okay to me. You don't need speed or angular velocity: curvature is a purely "geometric" property. There are formulas for finding curvature- do you know any?

If not what you are looking for is a circle passing through the same point, with the same tangent (first derivative) and same second derivative as this curve. Finding that will give you the radius of curvature and then the curvature.

I managed to come up with

$$\rho = \frac{{v^3 }}{{v_x g}} = \frac{{(v_x ^2 + v_y ^2 )^{3/2} }}{{v_x g}}$$

where vx/y is the initial velocity in the corrosponding direction. Does that look on track for the radius of curvature?

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Well, the orginal formula you gave for the trajectory didn't have vx or vy so I don't know.

However, one other thing you should think about- did you write the original formula correctly? I didn't notice when I replied before- I was assuming the trajectory must be a parabola- but what you wrote was
$$\overrightarrow r = (60t\cos 54)\widehat{\underline i } + (60t\sin 54 - 0.5g^2 )\widehat{\underline j }$$
That's linear in t: that trajectory is a straight line which has curvature 0 at all points! I assume that "-0.5 g2" was supposed to be "-0.5g t2".

HallsofIvy said:
Well, the orginal formula you gave for the trajectory didn't have vx or vy so I don't know.

However, one other thing you should think about- did you write the original formula correctly? I didn't notice when I replied before- I was assuming the trajectory must be a parabola- but what you wrote was
$$\overrightarrow r = (60t\cos 54)\widehat{\underline i } + (60t\sin 54 - 0.5g^2 )\widehat{\underline j }$$
That's linear in t: that trajectory is a straight line which has curvature 0 at all points! I assume that "-0.5 g2" was supposed to be "-0.5g t2".

Oops yep youre right. I missed a 't' there. It should be a parabola.

Anyway here's my final equation and evaluation for the radius of curvature:
$$\rho = \frac{{((v_0 \cos \theta )^2 + (v_o \sin \theta - gt)^2 )^{3/2} }}{{gv_0 \cos \theta }} = \frac{{[(60\cos 54)^2 + (60\sin 54 - 9.81(5.443))^2 ]^{3/2} }}{{(9.81)(60\cos 54)}} \approx 130.4m$$

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## What is the radius of curvature of a particle?

The radius of curvature of a particle is a measurement of how curved the path of a particle is at a specific point. It is defined as the radius of the circle that best approximates the curve of the particle's path at that point.

## How is the radius of curvature calculated?

The radius of curvature is calculated by taking the reciprocal of the curvature at a specific point. Curvature is calculated by finding the rate of change of the slope of the tangent line to the particle's path at that point.

## Why is the radius of curvature important in particle physics?

The radius of curvature is important in particle physics because it helps us understand the path of particles as they move through space. It can also give insights into the forces and interactions acting on the particle.

## What factors affect the radius of curvature of a particle?

The radius of curvature of a particle can be affected by several factors, including the particle's mass, velocity, and the strength of the forces acting on it. It can also be affected by the shape and curvature of the path the particle is traveling on.

## Can the radius of curvature of a particle change?

Yes, the radius of curvature of a particle can change as the particle moves through space and encounters different forces and interactions. It can also change if the particle's path changes, such as when it is deflected by a magnetic field or when it moves from one medium to another.