A person bends over, and lifts an object of mass m = 28.5 kg while keeping the back parallel with the floor. The muscle that attaches 2/3 of the way up the spine maintains the position of the back. This muscle is called the back muscle or latissimus dorsi muscle. The angle between the spine and the force
T
in this muscle is è = 10°. Use the balance of torques and forces in the simplified graph shown in part (b) of the attached figure and take the mass of the upper body as M = 42 kg.
Find the x-component of the compressive force
R
in the spine. Hint: In the figure, W1 is the weight of the object being lifted and W2 is the weight of the upper body.
To find the x-component of the compressive force R in the spine, we need to sum up all the torques acting on the body and set it equal to zero.
From the attached figure, we can see that there are three torques to consider:
1. The torque due to the weight of the object being lifted (W1) acting at a distance of 2/3 of the way up the spine, which can be represented as τ1 = W1 * d1.
2. The torque due to the weight of the upper body (W2) acting at a distance L/2, which can be represented as τ2 = W2 * L/2.
3. The torque due to the compressive force R acting at a distance L, which can be represented as τ3 = R * L.
Since the person is holding the object in equilibrium (not rotating), the sum of the torques must be zero:
τ1 + τ2 + τ3 = 0
Now, let's substitute the expressions for each torque:
(W1 * d1) + (W2 * L/2) + (R * L) = 0
To find the x-component of the compressive force R in the spine, we need to express the weight of the object and the weight of the upper body in terms of the gravitational force and the given masses:
W1 = m * g
W2 = M * g
Substituting these expressions into the torque equation:
(m * g * d1) + (M * g * L/2) + (R * L) = 0
To find the x-component of the compressive force R, we need to isolate it in the equation. Rearranging the equation:
R * L = -(m * g * d1) - (M * g * L/2)
R = (-(m * g * d1) - (M * g * L/2)) / L
Now, we can substitute the given values to get the final answer for the x-component of the compressive force R.