A soccer player extends her lower leg in a kicking motion by exerting a force with the muscle above the knee in the front of her leg. Suppose she produces an angular acceleration of 28.5 rad/s2 and her lower leg has a moment of inertia of 0.7 kg⋅m2. What is the force, in newtons, exerted by the muscle if its effective perpendicular lever arm is 2.05 cm?
.750(30)=F(1.90) f= 1184.2105N
To find the force exerted by the muscle, we use the equation:
Torque = Force * Lever arm
In this case, the torque is given by:
Torque = Moment of inertia * Angular acceleration
Let's calculate the torque first:
Torque = 0.7 kg⋅m^2 * 28.5 rad/s^2
= 19.95 N⋅m
We also convert the lever arm from centimeters to meters:
Lever arm = 2.05 cm * (1 m/100 cm)
= 0.0205 m
Now we can rearrange the equation to solve for force:
Force = Torque / Lever arm
Plugging in the values, we get:
Force = 19.95 N⋅m / 0.0205 m
= 971.22 N
Therefore, the force exerted by the muscle is approximately 971.22 newtons.
To find the force exerted by the muscle, we can use Newton's second law for rotational motion, which states that the torque (τ) applied to a body is equal to the product of the moment of inertia (I) and the angular acceleration (α):
τ = I * α
In this case, the torque exerted by the muscle is equal to the force applied multiplied by the perpendicular lever arm. The perpendicular lever arm is given in centimeters, but we need to convert it to meters before using it in the equation. So, the perpendicular lever arm (r) is:
r = 2.05 cm = 2.05 * 0.01 m = 0.0205 m
Now, we can rearrange the equation to solve for the force (F):
τ = F * r
F = τ / r
Substituting the given values, we have:
F = (I * α) / r
F = (0.7 kg⋅m^2) * (28.5 rad/s^2) / 0.0205 m
Calculating this expression, we get:
F ≈ 1018.29 N
Therefore, the force exerted by the muscle is approximately 1018.29 Newtons.