Question

Find torsion of the motion r(t)=⟨cos5t,sin5t,3t⟩

Answer 1

To find the torsion of the motion r(t) = ⟨cos 5t, sin 5t, 3t⟩, we can use the formula:

τ(t) = − det (r ′ (t), r ″ (t), r ‴ (t)) / ‖r ′ (t) × r ″ (t)‖2

First, we need to find the derivatives of r(t):

r ′ (t) = ⟨− 5 sin 5t, 5 cos 5t, 3⟩

r ″ (t) = ⟨− 25 cos 5t, − 25 sin 5t, 0⟩

r ‴ (t) = ⟨125 sin 5t, − 125 cos 5t, 0⟩

Then, we need to find the cross product of r ′ (t) and r ″ (t):

r ′ (t) × r ″ (t) = ⟨75 sin 5t, 75 cos 5t, 625⟩

Next, we need to find the determinant of the matrix formed by r ′ (t), r ″ (t), and r ‴ (t):

det (r ′ (t), r ″ (t), r ‴ (t)) = − 78125 sin 10t

Finally, we need to find the norm of r ′ (t) × r ″ (t):

‖r ′ (t) × r ″ (t)‖ = √(75^2 + 75^2 + 625^2) = 25√(2 + 2 + 50)

Therefore, the torsion of the motion is:

τ(t) = − (− 78125 sin 10t) / (25√(2 + 2 + 50))^2

τ(t) = 125 sin 10t / (2 + 2 + 50)

Answer 2

To determine the torsion of the given motion described as r(t) = ⟨cos5t, sin5t, 3t⟩, calculate the derivatives to find the tangent, normal, and binormal vectors, and use these to compute the torsion using vector calculus techniques.

Step-by-step explanation:

To find the torsion of the curve given by the vector function r(t) = ⟨cos5t, sin5t, 3t⟩, we must first calculate the curve's unit tangent vector T, the normal vector N, and the binormal vector B. The torsion τ is then determined using the formula τ = -(³r' × ²r'') · ³r'''.

First, calculate the first derivative ²r' to get the tangent vector, then normalize this to get the unit tangent vector T. Next, find the second derivative ³r'' and calculate the normal vector N by subtracting from ²r'' its projection onto T and normalize this to unit length. Then, find the third derivative ³r''' of r(t). We use T and N to calculate the binormal vector B as B = T × N. Finally, the torsion is given by τ = -B · ³r''', where · represents the dot product.

The steps outlined demonstrate the process of calculating torsion, which requires knowledge of vector calculus and involves differentiation and vector operations like cross and dot products.