Question

When you do a chin-up, you raise your chin just over a bar (the chinning bar), supporting yourself with only your arms. Typically, the body below the arms is raised by about 30 cm in a time of 1.0 s, starting from rest. Assume that the entire body of a 680-N person doing chinups is raised by 30 cm, and that half the 1.0 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 use it to find the force his arms must exert on him during the accelerating part of the chin-up.

Short answer

The arms must exert a force of 763.268 N during the upward acceleration.

Step-by-step solution

Understanding the Problem

In this problem, you perform a chin-up by raising your body 30 cm in 1 second. During this motion, you are accelerating upwards for the first half and accelerating downwards for the second half. We need to calculate the force exerted by the arms during the upward acceleration.

Draw the Free-Body Diagram

To analyze the forces, draw a free-body diagram. The key forces acting on the body are the gravitational force (weight), which is 680 N, and the force exerted by the arms (F_arm) that acts upwards. These are the two primary forces involved during the upward acceleration.

Calculate Acceleration

During the upward motion, the body accelerates uniformly for 0.5 seconds to cover the 30 cm distance. First, convert 30 cm to meters (0.30 m). Then, find the acceleration using the equation: \( d = \frac{1}{2} a t^2 \). Here, \( d = 0.15 \) m (half the distance during acceleration), and \( t = 0.5 \) s.

Solve for Acceleration

The equation becomes \( 0.15 = \frac{1}{2} a (0.5)^2 \). Solving for \( a \), we get:\[ 0.15 = 0.125a \]\[ a = \frac{0.15}{0.125} = 1.2 \, \text{m/s}^2 \]

Using Newton's Second Law

Apply Newton's second law during upward acceleration: \( F_{ ext{net}} = m a \). The net force is the sum of forces: \( F_{ ext{arm}} - F_{ ext{gravity}} = m a \). Here, \( F_{ ext{gravity}} = 680 \) N and \( m = \frac{680}{9.8} \) kg since weight = mass × gravity.

Calculate Mass

Calculate the mass of the person using the equation \( m = \frac{F_{ ext{gravity}}}{g} \).\[ m = \frac{680}{9.8} = 69.39 \, \text{kg} \]

Solve for Arm Force

Rearrange the equation \( F_{ ext{arm}} - 680 = 69.39 \times 1.2 \) to find \( F_{ ext{arm}} \).\[ F_{ ext{arm}} = 69.39 \times 1.2 + 680 \]\[ F_{ ext{arm}} = 83.268 + 680 \]\[ F_{ ext{arm}} = 763.268 \, \text{N} \]

Conclusion

The force that the arms must exert during the upward accelerating part of the chin-up is 763.268 N. This includes overcoming both the gravitational force and providing the necessary acceleration.

Key concepts

Free-Body Diagram

A free-body diagram is an essential tool for visualizing the forces acting on an object, in this case, a person doing a chin-up. To draw a free-body diagram:

1. Represent the object as a simple shape like a dot or a box. Here, it's the entire body of the person.
2. Identify and draw the forces acting on the body. For the person doing chin-ups:

• Draw an arrow pointing downward to represent the gravitational force (weight) of 680 N.
• Draw an arrow pointing upward to show the force exerted by the arms. This is the force needed to overcome gravity and cause upward acceleration.

The length of the arrows can represent the relative magnitude of each force. In the diagram, arrows help visualize the direction and size of the forces, clearly showing that the arm's force must be greater than the gravitational force to create upward acceleration.

Understanding free-body diagrams helps break down complex problems into simpler parts, making it easier to apply Newton's laws of motion and determine net forces.

Upward Acceleration

Upward acceleration refers to the increase in velocity that occurs when an object moves in an upward direction. In this chin-up exercise:

1. The person accelerates upwards during the first half of the 1-second motion.
2. The body needs to travel a total of 30 centimeters, which is split between upward and downward acceleration.

To calculate the acceleration during this upward motion, we first convert the distance into meters, resulting in 0.30 m total and 0.15 m for the upward phase. Using the equation:\[ d = \frac{1}{2} a t^2 \] we can set \( d = 0.15 \text{ m} \) and \( t = 0.5 \text{ s}\). Solving for \( a \), we get \[ 0.15 = \frac{1}{2} a (0.5)^2 \] which simplifies to \[ a = 1.2 \, \text{m/s}^2 \].

This upward acceleration is vital because it not only counteracts gravity but also provides the additional force needed to lift the body.

Calculating Force

Calculating the force required during a chin-up involves using Newton's Second Law. This law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:

\[ F_{\text{net}} = m a \]

In our chin-up example, there's a need to calculate the force exerted by the arms during the upward acceleration phase. First, we need to find mass using the weight of the person, given by gravity:

• Calculate mass \( m = \frac{F_{\text{gravity}}}{g} \), where \( F_{\text{gravity}} = 680 \, \text{N} \) and \( g = 9.8 \, \text{m/s}^2 \).
  Solving gives \( m = 69.39 \, \text{kg} \).

Now, calculate the arm force using Newton's law:

\[ F_{\text{arm}} - F_{\text{gravity}} = m a \]

Rearrange to find the arm force:
\[ F_{\text{arm}} = 69.39 \times 1.2 + 680 \]
\[ F_{\text{arm}} = 763.268 \, \text{N} \]

This force, 763.268 N, combines the amount needed to override the force of gravity with the additional force necessary to cause the body to accelerate upward effectively.