To study torque experimentally, you apply a force to a beam. One end of the beam is attached to a pivot that allows the beam to

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To study torque experimentally, you apply a force to a beam. One end of the beam is attached to a pivot that allows the beam to

rotate freely. The pivot is taken as the origin or your coordinate system. You apply a force of F = Fx i + Fy j + Fz k at a point r = rx i + ry j + rz k on the beam.Part (a) Enter a vector expression for the resulting torque, in terms of the unit vectors i, j, k and the components of F and r. Part (b) Calculate the magnitude of the torque, in newton meters, when the components of the position and force vectors have the values rx = 4.07 m, ry = 0.075 m, rz = 0.035 m, Fx = 2.8 N, Fy = 8.4 N, Fz = 1.4 N. Part (c) If the moment of inertia of the beam with respect to the pivot is I = 241 kg˙m², calculate the magnitude of the angular acceleration of the beam about the pivot, in radians per second squared.

Physics

1 answer:

kicyunya [3.2K] · Answer 1 · 2026-03-20T04:49:58+03:00

Answer:

a) τ = i ^ (y $F_{z}$ - z $F_{y}$ ) + j ^ (z Fₓ - x $F_{z}$ ) + k ^ (x $F_{y}$ - y Fₓ)

b) τ = (-0.189i ^ -5.6 j ^ + 33.978k ^) N m

c) α = (-7.8 10⁻⁴ i ^ - 2.3 10⁻² j ^ + 1.4 10⁻¹ k ^) rad / s²

Explanation:

a) The torque can be expressed as

τ = r x F

To tackle this equation, using the determinant approach is the most straightforward method

$\tau =\left[\begin{array}{ccc}i&j&k\\x&y&z\\F_{x}&F_{y} &F_{z}\end{array}\right]$

The resulting expression is

τ = i ^ (y $F_{z}$ - z $F_{y}$ ) + j ^ (z Fₓ - x $F_{z}$ ) + k ^ (x $F_{y}$ - y Fₓ)

b) Now let's compute

τ = i ^ (0.075 1.4 -0.035 8.4) + j ^ (0.035 2.8 - 4.07 1.4) + k ^ (4.07 8.4 - 0.075 2.8)

τ = i ^ (- 0.189) + j ^ (-5.6) + k ^ (33,978)

τ = (-0.189i ^ -5.6 j ^ + 33.978k ^) N m

c) To find angular acceleration, we use

τ = I α

α = τ / I

The moment of inertia being a scalar means that only the magnitude of each component changes, orientation remains constant.

α = (-0.189i^ -5.6 j^ + 33.978k^) / 241

α = (-7.8 10⁻⁴ i ^ - 2.3 10⁻² j ^ + 1.4 10⁻¹ k ^) rad / s²