Q9: Prb 5 - This problem has two parts to this. I will type out the questions for each of the two parts along with additional info in "Description" below.

Question

Consider a person doing push-ups. The person has a mass of 58.9 kg, and the distance from their feet to their center of mass is 0.98 m, while the distance from their feet to their hands is 1.55 m.

Part a) What force in newtons should the person exert on the floor with each hand to do a push-up? Assume that this person moves up at a constant speed.

Part b) The triceps muscle at the back of their upper arm has an effective lever arm of 1.7 cm, and this person exerts force on the floor at a horizontal distance of 18.5 cm from the elbow joint. Calculate the magnitude of the force in newtons for each triceps muscle.

Answer 1

Let's break down the problem into two parts:

Part a: Force on the Hands

To do a push-up at a constant speed, the person's net force upward (provided by the hands) must equal their weight (mg) downward. Here, m is the mass of the person (58.9 kg), and g is the acceleration due to gravity (approximately 9.81 m/s²).

F_net = m * g

F_net = 58.9 kg * 9.81 m/s² ≈ 578.049 N

Since the person is exerting this force with both hands, the force exerted on each hand is:

F_hand = F_net / 2

F_hand = 578.049 N / 2 ≈ 289.025 N

So, each hand should exert a force of approximately 289.025 newtons on the floor to do a push-up at a constant speed.

Part b: Force on the Triceps Muscle

The force exerted by the triceps muscle can be calculated using the concept of torque. Torque (τ) is the product of the force applied and the lever arm distance:

τ = Force × Lever Arm

First, convert the lever arm from centimeters to meters:

Lever Arm = 1.7 cm = 0.017 m

Now, you mentioned that the person exerts force on the floor at a horizontal distance of 18.5 cm from the elbow joint. Convert this to meters as well:

Distance from elbow joint = 18.5 cm = 0.185 m

Now, calculate the force exerted by the triceps muscle:

τ = Force_triceps × Lever Arm

τ = Force_triceps × 0.017 m

To maintain equilibrium, the torque provided by the triceps muscle must counteract the torque due to the weight of the person's arm:

τ_triceps = τ_weight_of_arm

Force_triceps × 0.017 m = (mass_of_arm × g) × 0.185 m

We can solve for Force_triceps:

Force_triceps = (mass_of_arm × g) × (0.185 m / 0.017 m)

Now, we need to calculate the mass of the person's arm. Assuming a constant arm density and a cylindrical shape:

Mass_of_arm = (π * r² * L) * density

Where:

1. r is the radius of the arm (considering it as a cylinder)
2. L is the length of the arm from the elbow to the hand
3. density is the density of human tissue

You would need to estimate these values based on typical human body proportions. Once you have the mass_of_arm, you can plug it into the equation for Force_triceps to calculate the force exerted by the triceps muscle.