Problem 61 - University Physics with Modern Physics

Question

People who do chinups raise their chin just over a bar (the chinning bar), supporting themselves with only their arms. Typically, the body below the arms is raised by about 30 \(\mathrm{cm}\) in a time of 1.0s, starting from rest. Assume that the entire body of a \(680-\mathrm{N}\) person doing chin-ups 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. Draw a free-body diagram 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 force exerted by the arms is approximately 763 N.

Step 1: Understand the Problem

The problem requires us to find the force exerted by a person's arms while doing a chin-up. The person weighs 680 N and raises their body by 30 cm in 1 second. This motion includes two phases: accelerating upward for 0.5 seconds and accelerating downward for 0.5 seconds.

Step 2: Setup the Free-Body Diagram

In the free-body diagram of the person during the upward motion, the forces acting are: the gravitational force pulling downward, denoted as \( F_g = 680 \) N, and the force exerted by the arms upward, denoted as \( F_a \). During upward acceleration, \( F_a \) must be greater than \( F_g \) for the person to accelerate upwards.

Step 3: Calculate the Acceleration

The person accelerates uniformly to a maximum height in 0.5 seconds. The total distance covered during upward acceleration is 15 cm (half of 30 cm). Using the equation of motion for uniformly accelerated motion, \( s = \frac{1}{2} a t^2 \), where \( s = 0.15 \) m and \( t = 0.5 \) s, solve for \( a \).

Step 4: Solve for Acceleration

Rearrange the motion equation to find \( a \): \[ a = \frac{2s}{t^2} = \frac{2 \times 0.15}{(0.5)^2} = 1.2 \, \text{m/s}^2 \].

Step 5: Apply Newton's Second Law

Using Newton's second law, \( F = ma \), find the net force needed. Convert the person's weight to mass: \( m = \frac{F_g}{g} = \frac{680}{9.8} \approx 69.39 \) kg. The net force during upward acceleration is \( F_{net} = ma = 69.39 \times 1.2 \approx 83.27 \) N.

Step 6: Calculate Force Exerted by the Arms

The arms must exert a force equal to the gravitational force plus the net force needed to accelerate upward: \[ F_a = F_g + F_{net} = 680 + 83.27 = 763.27 \, \text{N} \].

Key Concepts

Free-Body Diagram

When working with problems involving forces, a free-body diagram is an essential tool. It helps visualize the forces acting on an object, in this case, a person doing chin-ups. The diagram simplifies complex problems by representing the body as a point and using arrows to illustrate forces.
This simple graphical representation helps us concentrate on the critical forces at play:

• Gravitational force (weight) acts downward and is represented by an arrow pointing towards the earth.
• Force exerted by the arms acts upward to counteract gravity and lift the person, shown by an arrow pointing upwards.

For a person weighing 680 N, the gravitational force is also 680 N acting downward. To lift the person during chin-ups, the upward force exerted by the arms must be greater than this gravitational pull during the upward acceleration.

Uniform Acceleration

Uniform acceleration means that the change in velocity is constant over time. In the chin-up exercise, this occurs in two phases: upward and downward, each lasting 0.5 seconds. Uniform acceleration simplifies calculations because it implies there's a consistent rate of speeding up or slowing down.
During the upward phase, the person's body gains velocity due to a constant upward acceleration. Likewise, in the downward phase, the deceleration is uniform, bringing the body back to rest over the same period. This consistency allows us to use straightforward kinematic equations to determine acceleration values required for further calculations.

• For distance upward: uniform acceleration over 0.5 seconds covers 15 cm.
• Equation: \(s = \frac{1}{2} a t^2\) helps solve for acceleration \(a\).

These principles of uniform motion are crucial for accurately calculating net forces and ultimately the force exerted by the arms.

Force Calculation

Calculating forces involves applying Newton's Second Law, which states that force equals mass times acceleration. The arm's force must overcome both gravity and provide additional force for upward acceleration.
First, determine the mass of the person using gravitational force: \[ m = \frac{F_g}{g} = \frac{680}{9.8} \approx 69.39 \, \text{kg} \]Then, calculate the net force needed for upward motion using the previously calculated acceleration: \[ F_{net} = ma = 69.39 \times 1.2 \approx 83.27 \, \text{N} \]Finally, add this net force to the gravitational force to find the total force exerted by the arms: \[ F_a = F_g + F_{net} = 680 + 83.27 = 763.27 \, \text{N} \]These calculations ensure that the arms apply sufficient force to lift the body despite opposing gravitational forces.

Gravitational Force

Gravitational force is the pull exerted by the earth on any object with mass. It is calculated by multiplying the mass of the object by the acceleration due to gravity, typically approximated as \(9.8 \, \text{m/s}^2\) on the surface of the Earth.
In the context of a chin-up, gravitational force acts downward and is equivalent to the weight of the person. For our scenario, the person has a gravitational force of 680 N, meaning that's the weight acting downward.

• Weight = Mass \(\times\) Gravity: \( F_g = mg \)
• For a 680 N force, mass is about 69.39 kg, using \( g = 9.8 \, \text{m/s}^2\).

Understanding gravitational force is crucial as it plays a significant role in determining the effort required by the arms to lift the body against this persistent downward pull.