Question

Force to Hold a Baseball A person holds a \(1.42-\mathrm{N}\) baseball in his hand, a distance of \(34.0 \mathrm{~cm}\) from the elbow joint, as shown in Figure \(8.30\). The biceps, attached at a distance of \(2.75 \mathrm{~cm}\) from the elbow, exerts an upward force of \(12.6 \mathrm{~N}\) on the forearm. Consider the forearm and hand to be a uniform rod with a mass of \(1.20 \mathrm{~kg}\). (a) Calculate the net torque acting on the forearm and hand. Use the elbow joint as the axis of rotation. (b) If the net torque obtained in part (a) is nonzero, in which direction will the forearm and hand rotate?

Short answer

Net torque is \(-2.13754 \mathrm{~Nm}\), causing clockwise rotation.

Step-by-step solution

Identify Forces and Distances

First, identify the forces acting on the system. The biceps exert an upward force of \(12.6 \mathrm{~N}\) at a distance of \(2.75 \mathrm{~cm}\) from the elbow. The baseball exerts a downward force of \(1.42 \mathrm{~N}\) located \(34.0 \mathrm{~cm}\) from the elbow. The gravitational force on the forearm and hand acts downward and can be calculated by \(1.20 \mathrm{~kg} \times 9.81 \mathrm{~m/s^2}\) and its point of action is the center of the forearm-hand system, which is half its length in this model.

Calculate Gravitational Force on the Forearm and Hand

Calculate the force due to gravity acting on the mass of the forearm and hand assuming a uniform rod. \(F_g = 1.20 \mathrm{~kg} \times 9.81 \mathrm{~m/s^2} = 11.772 \mathrm{~N}\). This force acts downward at the center, which we'll assume is \(17.0 \mathrm{~cm} \) from the elbow (half of the \(34.0 \mathrm{~cm}\)).

Convert Distances to Meters

To ensure proper calculation, convert all distances from centimeters to meters: \(34.0 \mathrm{~cm} = 0.34 \mathrm{~m}\), \(2.75 \mathrm{~cm} = 0.0275 \mathrm{~m}\), and the center of mass distance \(17.0 \mathrm{~cm} = 0.17 \mathrm{~m}\).

Calculate Torque for Each Force

Calculate the torque due to each force about the elbow joint:1. **Torque by Baseball**: \( \tau_1 = 1.42 \mathrm{~N} \times 0.34 \mathrm{~m} = 0.4828 \mathrm{~Nm} \) (clockwise).2. **Torque by Biceps**: \( \tau_2 = 12.6 \mathrm{~N} \times 0.0275 \mathrm{~m} = 0.3465 \mathrm{~Nm} \) (counterclockwise).3. **Torque by Forearm/Hand Mass**: \( \tau_3 = 11.772 \mathrm{~N} \times 0.17 \mathrm{~m} = 2.00124 \mathrm{~Nm} \) (clockwise).

Calculate Net Torque

Combine the torques to determine the net torque on the system: \[ \tau_{net} = -0.4828 - 2.00124 + 0.3465 = -2.13754 \mathrm{~Nm} \]The negative sign indicates that the net torque is clockwise.

Determine Direction of Rotation

Since the net torque is negative, the forearm and hand will rotate in the clockwise direction if not balanced by other forces or actions.

Key concepts

Net Torque

Understanding net torque is crucial when analyzing the rotational motion of objects. Torque, also known as the moment of force, is the measure of the turning effect produced by a force acting at a distance from a pivot point or axis of rotation. To calculate net torque, you need to consider the combination of all individual torques acting on a system, taking into account their directions.

• Torque is calculated as the product of force and the perpendicular distance from the axis of rotation to the line of action of the force.
• The direction of torque can either be clockwise (negative) or counterclockwise (positive), depending on the direction of the force applied.
• Sum up all torques acting on the system to find the net torque; if these are balanced, the net torque is zero, meaning no new rotation is induced.

In our given exercise, we identified three separate torques caused by different forces acting at varying distances from the elbow joint. By summing these, we calculated the net torque, which turned out to be clockwise, indicating a tendency for rotation in this direction.

Forces and Distances

In torque calculations, the concepts of force and distance play essential roles. The exerted force and its distance from the pivot point determine the torque magnitude produced. It's the perpendicular distance between the force application point and the rotation axis that counts most.

• The longer the distance, the greater the torque for the same force, due to increased leverage.
• In our example, the forces include the upward force by the biceps and the downward forces from the baseball and gravitational pull on the forearm/hand system.
• Each force was multiplied by its respective distance from the elbow joint (the axis) to calculate its specific torque.

Understanding how forces at different distances contribute to torque helps in predicting the resulting rotation motion of the system.

Elbow Joint

The elbow joint serves as the axis of rotation in the exercise, and it is a crucial reference point for calculating torque. Viewing it this way helps us understand how our body naturally influences movement. In context, the elbow acts like the pivot around which the forearm rotates. This aids in assessing the impacts of different forces exerted at various places along the arm.

• The elbow joint's location determines the distance components used in torque calculations.
• It acts as a key point where forces by muscle action (like the biceps) and external loads (such as holding an object) interact to affect arm movement.
• This understanding provides insights into biomechanical processes and helps in designing supportive aids or therapeutic exercises.

Recognizing the elbow joint's role and its relation to external forces is fundamental to comprehending the biomechanical behavior of arms.

Rotational Motion

Rotational motion deals with the motion of a body around a fixed point or axis. In this context, we're looking at how various forces contribute to the arm's rotational movement about the elbow joint. Understanding rotational motion involves several principles:

• Any imbalance in forces or torques leads to rotation or acceleration around the pivot, like the elbow.
• The moment of inertia, akin to mass in linear motion, affects how easily an object spins around an axis.
• In this problem, the net torque was found to be clockwise, hinting that the motion created would drive the arm in a downward rotation if unimpeded.

Analyzing rotational motion helps us predict not just how objects will move, but also guides us in applying forces to control or utilize this motion effectively, particularly in biomechanics and physical activities.