Suppose the hand shown in (Figure 1) holds an 8.9-kg mass. The total mass of the arm is 3.3 kg. What force, FM, is required of the deltoid muscle, assuming the mass is 52 cm from the shoulder joint?
So ive tried this problem multiple times but i'm not getting the correct answer,
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To find the force required from the deltoid muscle, we can use Newton's second law, which states that the force (F) is equal to the mass (m) multiplied by the acceleration (a).
In this case, the deltoid muscle is responsible for lifting the mass, so the force required is the force to lift the mass against gravity.
Step 1: Convert the distance from the shoulder joint from centimeters (cm) to meters (m).
52 cm = 52/100 = 0.52 m
Step 2: Calculate the weight of the mass.
Weight = mass * acceleration due to gravity
Weight = 8.9 kg * 9.8 m/s^2 ≈ 87.22 N
Step 3: Calculate the torque exerted by the weight of the mass about the shoulder joint.
Torque = weight * distance
Torque = 87.22 N * 0.52 m ≈ 45.37 Nm
Step 4: The deltoid muscle must exert a force to counteract this torque. Since the moment arm of the deltoid muscle is perpendicular to the arm's length, we can assume it is equal to the distance from the shoulder joint to the center of mass.
Torque = force * moment arm
45.37 Nm = FM * 0.52 m
Step 5: Solve for the force required from the deltoid muscle.
FM = 45.37 Nm / 0.52 m ≈ 87.25 N
Therefore, the force required from the deltoid muscle, FM, is approximately 87.25 N.
To solve this problem, you need to understand the concept of torque and its relationship with force and distance. Torque is the rotational equivalent of force and is calculated by multiplying the force applied to an object by the distance from the axis of rotation.
First, let's calculate the torque exerted by the weight of the mass. The formula for torque is:
Torque = Force x Distance
Here, the force is the weight of the mass (8.9 kg) multiplied by the acceleration due to gravity (9.8 m/s^2). The distance is given as 52 cm, but it is more convenient to convert it to meters:
Distance = 52 cm = 0.52 m
Thus, the torque exerted by the weight of the mass is:
Torque_mass = (8.9 kg) x (9.8 m/s^2) x (0.52 m)
Next, let's calculate the torque exerted by the arm. This torque is created by the weight of the arm acting at its center of mass. The force is the weight of the arm (3.3 kg) multiplied by the acceleration due to gravity (9.8 m/s^2). The distance is given as 52 cm, which we will convert to meters:
Distance = 52 cm = 0.52 m
Thus, the torque exerted by the arm is:
Torque_arm = (3.3 kg) x (9.8 m/s^2) x (0.52 m)
To find the force required of the deltoid muscle, we need to subtract the torque exerted by the arm from the torque exerted by the weight of the mass:
FM = Torque_mass - Torque_arm
Now you can plug in the values and calculate the force required:
FM = (8.9 kg x 9.8 m/s^2 x 0.52 m) - (3.3 kg x 9.8 m/s^2 x 0.52 m)
Calculating this expression will give you the force required of the deltoid muscle (FM).
Make sure you double-check your calculations to ensure accuracy. If you are still having trouble getting the correct answer, check the unit conversions and be mindful of any potential rounding errors.