Problem 18 - College Physics

Question

\(\bullet$$\bullet\) Forces during chin-ups. People who do chin-ups raise their chin just over a bar (the chinning bar), supporting themselves only by their arms. Typically, the body below the arms is raised by about 30 \(\mathrm{cm}\) in a time of 1.0 s, starting from rest. Assume that the entire body of a 680 N person who is chinning is raised this distance and that half the 1.0 \(\mathrm{s}\) is spent accelerating upward and the other half accelerating downward, uniformly in both cases. Make a free-body dia- gram of the person's body, and then apply it to find the force his arms must exert on him during the accelerating part of the chin-up.

Step-by-Step Solution

Answer

The arms exert 846.54 N during acceleration.

Step 1: Understand the Problem and Draw the Free-Body Diagram

The person performing a chin-up raises their body by 30 cm in 1 second. The problem involves uniform acceleration while going up and down. Begin by identifying forces: the gravitational force acts downward, and the force exerted by the arms acts upward. The free-body diagram of the person during upward acceleration consists of the weight of their body \( 680 \, \text{N}\) acting downward and the force from the arms \( F \, \text{N}\) acting upward.

Step 2: Calculate the Required Acceleration

We first need to calculate the acceleration experienced by the person. The time spent accelerating upward is \( t = 0.5 \, \text{s} \). Since the person starts from rest and covers a distance \( s = 0.3 \, \text{m} \) under uniform acceleration, use the equation: \[ s = \frac{1}{2} a t^2 \] Solve for acceleration \( a \): \[ 0.3 = \frac{1}{2} a (0.5)^2 \] \[ a = \frac{0.3 \times 2}{0.25} = 2.4 \, \text{m/s}^2 \]

Step 3: Apply Newton's Second Law

Apply Newton's second law \( F_{net} = ma \) during the upward motion. The net force \( F_{net} \) is the difference between the upward force exerted by the arms and the weight of the body: \[ F - 680 = \frac{680}{9.8} \times 2.4 \] Simplifying, we find the upward force \( F \). First, calculate the mass: \[ m = \frac{680}{9.8} \approx 69.39 \, \text{kg} \] Then plug into the equation: \[ F - 680 = 69.39 \times 2.4 \] \[ F = 166.54 + 680 \] \[ F = 846.54 \, \text{N} \]

Step 4: Verify Your Solution

Verify that the solution makes sense by reviewing the problem conditions. The force calculated (846.54 N) is greater than the body weight (680 N) because it needs to include the additional force necessary to accelerate the body upward.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation that depicts all the forces acting on an object. When a person performs a chin-up, they are fighting against gravity to lift their body upward. In the free-body diagram for this case, the forces need to be clearly identified:

• Gravitational Force: This acts downward and is equal to the weight of the person. For a person weighing 680 N, the gravitational force is also 680 N directed towards the Earth.
• Force Exerted by Arms: This is the upward force exerted by the muscles in the arms during the chin-up. It counteracts the gravitational force to raise the body.

The combination of these forces helps us understand the mechanics of the chin-up motion. During the initial phase when the person is accelerating upward, the force exerted by the arms must be greater than the gravitational force to overcome it and allow the upward movement of the body.

Uniform Acceleration

Uniform acceleration occurs when the rate of velocity change of an object is consistent over time. In the chin-up exercise, the person's body experiences uniform acceleration both while moving upward and returning downward. The process can be broken down as follows:

• Upward Motion: The person starts from rest, meaning their initial velocity is 0 m/s. They accelerate uniformly to reach a height of 30 cm in 0.5 seconds.
• Equation of Motion: The distance covered under uniform acceleration is calculated using the equation:\[ s = \frac{1}{2} a t^2 \]Here, \( s \) is the displacement (0.3 m), \( a \) is the acceleration, and \( t \) is the time period (0.5 s).
• Calculation: Solving for \( a \) using the values provided, we find that the acceleration is 2.4 m/s².

This orderly acceleration ensures the person's upward velocity increases uniformly until they reach the peak of the chin-up motion, at which point they start decelerating or accelerating downward uniformly.

Force Calculation

The force calculation in this exercise relies on Newton's Second Law of Motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)). Here's how it's applied in our chin-up scenario:

• Mass Calculation: First, determine the mass of the person by dividing their weight by the gravitational acceleration (\( 680 \, \text{N} / 9.8 \, \text{m/s}^2 \approx 69.39 \, \text{kg} \)).
• Net Force Equation: During the upward phase of the chin-up, the net force \( F_{net} \) is the total force exerted by the arms minus the gravitational force:\[ F - 680 = 69.39 \times 2.4 \]
• Solving for Upward Force: Rearrange the equation to solve for \( F \):\[ F = 166.54 + 680 = 846.54 \, \text{N} \]This upward force of 846.54 N exceeds the gravitational pull of 680 N, demonstrating that extra force was needed to accelerate the body upwards.

By following these calculations, the force required for the chin-up's acceleration phase is determined, illustrating how the arms facilitate lifting the body against gravity.