The drawing shows a model for the motion of the human forearm in throwing a dart. Because of the force applied by the triceps muscle, the forearm can rotate about an axis at the elbow joint. Assume that the forearm has the dimensions shown in the drawing and a moment of inertia of 0.080 kg · m2 (including the effect of the dart) relative to the axis at the elbow. Assume also that the force acts perpendicular to the forearm. Ignoring the effect of gravity and any frictional forces, determine the magnitude of the force needed to give the dart a tangential speed of 5.2 m/s in 0.11 s, starting from rest.
460
To determine the magnitude of the force needed to give the dart a tangential speed of 5.2 m/s in 0.11 s, starting from rest, we can use the principle of conservation of angular momentum.
Angular momentum is given by the formula:
L = Iω
Where:
L = angular momentum
I = moment of inertia
ω = angular velocity
Since the forearm starts from rest, the initial angular velocity is 0. Therefore, the change in angular momentum can be calculated as:
ΔL = Lf - Li = Iωf - Iωi = I(ωf - ωi)
The change in angular momentum is equal to the torque applied to the forearm multiplied by the time it acts:
ΔL = τΔt
The torque can be calculated as:
τ = Fr
Where:
τ = torque
F = force
r = distance from the axis of rotation (in this case, the elbow joint)
In this problem, the force acts perpendicular to the forearm, so it is responsible for causing the rotation. The distance from the axis of rotation is the length of the forearm, which is given in the drawing.
Substituting the values into the equation, we have:
I(ωf - ωi) = FrΔt
Now, let's solve for the force (F):
F = (I(ωf - ωi)) / (rΔt)
Substituting the given values:
F = (0.080 kg · m^2) * ((5.2 m/s - 0 m/s) / (0.31 m) * (0.11 s)
Calculating this expression will give us the magnitude of the force needed to give the dart a tangential speed of 5.2 m/s in 0.11 s, starting from rest.