Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

Problem 38a

Chapter 10, Problem 38a

The forearm in Fig. 10–57 accelerates a 3.6-kg ball at 7.0 m/s² by means of the triceps muscle, as shown. Calculate the torque needed.

Verified step by step guidance

Identify the key variables in the problem: the mass of the ball (m = 3.6 kg), the acceleration of the ball (a = 7.0 m/s²), and the lever arm distance (r). The lever arm distance is typically provided in the diagram, but since the image is not available, we will denote it as r for now.

Calculate the force exerted by the ball using Newton's second law: \( F = m \cdot a \). Substitute the given values for mass and acceleration into the formula.

Understand that torque (\( \tau \)) is the product of the force (F) and the lever arm distance (r), and is given by the formula: \( \tau = F \cdot r \cdot \sin(\theta) \), where \( \theta \) is the angle between the force vector and the lever arm. Assume \( \theta \) is 90° unless otherwise specified, which simplifies \( \sin(\theta) \) to 1.

Substitute the calculated force (F) and the lever arm distance (r) into the torque formula: \( \tau = F \cdot r \). Ensure the units are consistent (e.g., meters for r).

Conclude by noting that the torque depends on the value of r, which should be obtained from the diagram or problem context. If r is provided, substitute it to compute the numerical value of the torque.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). It is essential in understanding how forces cause objects to rotate around an axis. The direction of torque is determined by the right-hand rule, and it plays a crucial role in analyzing the mechanics of the forearm and muscle actions.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is fundamental in calculating the force required to accelerate the 3.6-kg ball at 7.0 m/s². Understanding this law helps in determining the necessary force exerted by the triceps muscle to achieve the desired acceleration.

Lever Arm

The lever arm is the perpendicular distance from the line of action of the force to the pivot point of rotation. In the context of the forearm, it is crucial for calculating torque, as a longer lever arm results in greater torque for the same amount of force. This concept is vital for understanding how muscles generate torque to lift or accelerate objects.

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