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 31b

Chapter 10, Problem 31b

The bolts on the cylinder head of an engine require tightening to a torque of 95 m-N. If the six-sided bolt head is 15 mm across (Fig. 10–55), estimate the force applied near each of the six points by a wrench.

Verified step by step guidance

Understand the relationship between torque, force, and lever arm. Torque (τ) is given by the formula:

τ = F r sin (θ)

, where F is the force applied, r is the lever arm (distance from the axis of rotation to the point of force application), and θ is the angle between the force and the lever arm.

In this problem, the force is applied perpendicular to the lever arm, so

\sin (θ) = 1

. This simplifies the torque formula to:

τ = F r

The lever arm (r) is half the distance across the bolt head, as the force is applied near the edge. Convert the given distance of 15 mm to meters:

r = \frac{15}{2} \times 10 ⁻ 3 = 0.0075 m

Rearrange the torque formula to solve for the force:

F = \frac{τ}{r}

. Substitute the given torque (95 N·m) and the calculated lever arm (0.0075 m) into the equation.

Perform the division to find the force applied near each of the six points. This force will be the same at each point since the torque is distributed evenly across the bolt head.

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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). In this case, the torque required to tighten the bolts is given as 95 m•N, indicating the amount of rotational force needed to achieve the desired tightness.

Lever Arm

The lever arm is the perpendicular distance from the line of action of the force to the pivot point. For the bolts, the lever arm is half the width of the bolt head, which is 7.5 mm (or 0.0075 m). This distance is crucial for calculating the force applied, as it directly influences the torque generated by the wrench.

Force Distribution

When multiple points of force are applied, such as with the six bolts, the total torque must be distributed among them. To find the force at each bolt, the total torque is divided by the number of bolts and the lever arm. This concept is essential for understanding how the torque requirement translates into the force exerted at each bolt head.

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Related Practice

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