A 500 g steel block rotates on a steel table while attached to a 1.2-m-long hollow tube as shown in FIGURE CP8.70. Compressed air fed through the tube and ejected from a nozzle on the back of the block exerts a thrust force of 4.0 N perpendicular to the tube. The maximum tension the tube can withstand without breaking is 50 N. If the block starts from rest, how many revolutions does it make before the tube breaks?

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Convert the mass of the steel block from grams to kilograms. Since 1 g = 0.001 kg, the mass of the block is $ m = 0.5 \; \text{kg} $.

Determine the net force acting on the block. The thrust force $ F_t = 4.0 \; \text{N} $ acts perpendicular to the tube, providing the centripetal force necessary for circular motion. The tension in the tube will increase as the block accelerates.

Use Newton's second law for rotational motion to relate the centripetal force $ F_c $ to the tension in the tube. The centripetal force is given by $ F_c = \frac{m v^2}{r} $, where $ v $ is the tangential velocity and $ r = 1.2 \; \text{m} $ is the radius of the circular path.

Calculate the work done by the thrust force to increase the kinetic energy of the block. The work-energy principle states that $ W = \Delta KE $, where $ W = F_t \cdot d $ and $ \Delta KE = \frac{1}{2} m v^2 $. Here, $ d $ is the distance traveled by the block along the circular path, which can be expressed as $ d = 2 \pi r \cdot n $, where $ n $ is the number of revolutions.

Set the maximum tension $ T_{max} = 50 \; \text{N} $ equal to the centripetal force $ F_c $ and solve for the tangential velocity $ v $. Then, substitute $ v $ into the work-energy equation to solve for $ n $, the number of revolutions before the tube breaks.

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Key Concepts

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

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 crucial for understanding how the thrust force from the compressed air affects the motion of the steel block, allowing us to calculate the resulting acceleration and subsequent motion.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. In this scenario, the thrust force contributes to the centripetal force needed to maintain the block's circular motion, and understanding this relationship is essential for determining how long the block can rotate before the tube breaks.

Tension in the Tube

Tension refers to the force transmitted through the tube when the block rotates. The maximum tension the tube can withstand (50 N) is a critical factor in this problem, as it sets the limit for the forces acting on the block. By analyzing the forces involved, we can determine how many revolutions the block can make before reaching this breaking point.

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