Problem 16 - College Physics

Question

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 \(1.0 \mathrm{~s},\) starting from rest. Assume that the entire body of a \(680 \mathrm{~N}\) 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 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 958 N.

Step 1: Understand the Problem

When doing a chin-up, the person lifts their body by using the force from their arms to overcome gravitational force, causing the body to accelerate upwards.

Step 2: Draw a Free-Body Diagram

Draw the free-body diagram of the person doing the chin-up. Identify two forces: the gravitational force downward ( F_g = 680 ext{ N} ) and the upward force exerted by the arms ( F_a ).

Step 3: Analyze Motion and Forces

Since the person accelerates upwards and decelerates back to rest in equal time intervals, we can analyze the upward acceleration period separately. The gravitational force remains constant at 680 N, while the arm force needs to be enough to not only support this weight but also provide additional force for upward acceleration.

Step 4: Calculate Acceleration

Using the kinematic equations, and knowing the distance (s = 0.30 m) and time (t = 0.5 s for acceleration), calculate the acceleration (ag) using the formula: \[ s = \frac{1}{2} a (\frac{t}{2})^2 \] Solving this results in: a = 4\, ext{m/s}^2\.

Step 5: Apply Newton's Second Law

Using Newton's second law, \[ F = ma \] we can set up the equation: \[ F_a - F_g = m \cdot a \] where m = F_g / g = 680 / 9.8 kg. By substituting as value:\[ F_a = (\frac{680}{9.8}) \times 4 + 680 \]

Step 6: Solve for the Arm Force

Plug in the values to find the force exerted by the arms:\[ F_a = (69.4 \times 4) + 680 = 957.6\, ext{N} \]

Key Concepts

Newton's Second Law

Newton's Second Law is central to understanding motion in physics. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This fundamental law can be expressed through the equation \( F = ma \), where \( F \) is the net force applied, \( m \) is the mass of the object, and \( a \) is its acceleration.
The law allows us to determine the relationship between forces and motion. In the context of the chin-up problem, Newton's Second Law helps us determine the force required by the arms to lift the body. As the body of the person accelerates upward during the chin-up, applying \( F = ma \) enables us to calculate how much additional force, beyond the gravitational force, is needed.

Free-Body Diagram

A free-body diagram is a useful tool in physics for visualizing the forces acting on an object. It illustrates all of the external forces to better analyze a situation. This tool helps clearly identify and separate forces to simplify complex interactions.
For the chin-up problem, the free-body diagram includes:

• Gravitational force \(F_g\), which is 680 N, pulling the individual downward.
• Force exerted by the arms \(F_a\), which acts upward opposing the gravitational pull.

This simplification is crucial because it allows us to focus on the forces that directly impact motion. By accurately depicting the forces involved, the diagram aids in effectively applying physics laws to solve for unknowns, such as the necessary force the arms must exert.

Kinematics

Kinematics is a branch of physics that deals with motion without considering the forces that cause it. It describes how objects move through equations that relate distances, speeds, and accelerations over time.
In the chin-up scenario, we rely on kinematic equations to determine the acceleration of the person as they lift their body. Given the information:

• Displacement is 0.30 m during the upward motion.
• Time spent accelerating is 0.5 s.

This allows using the equation \( s = \frac{1}{2} a t^2 \) to solve for the acceleration \( a \), resulting in an acceleration value of 4 m/s². This derived acceleration is vital for the subsequent force calculations using Newton's Second Law.

Forces in Physics

Understanding forces is key to analyzing physical movements. Forces are pushes or pulls that can cause objects to start moving, stop, or change their motion. In physics, forces are vectors with both magnitude and direction.
In the chin-up example, two primary forces are at play:

• Gravitational force, which pulls the chinning individual's body downwards at 680 N.
• The exerted force by the arms required to lift the body upward.

These forces must be examined in tandem to understand how they impact motion. The arm force is particularly important, as it must not only counteract gravity but also provide sufficient energy for upward movement. Solving for this additional force is crucial to completing the chin-up, highlighting how forces interact to influence physical activity.