Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 54

Chapter 8, Problem 54

FIGURE P8.54 shows two small 1.0 kg masses connected by massless but rigid 1.0-m-long rods. What is the tension in the rod that connects to the pivot if the masses rotate at 30 rpm in a horizontal circle?

Verified step by step guidance

Step 1: Convert the rotational speed from revolutions per minute (rpm) to angular velocity in radians per second. Use the formula $ \omega = \frac{2\pi \times \text{rpm}}{60} $.

Step 2: Determine the centripetal force acting on each mass due to its circular motion. The centripetal force is given by $ F_c = m \cdot \omega^2 \cdot r $, where $ m $ is the mass, $ \omega $ is the angular velocity, and $ r $ is the radius of rotation.

Step 3: Analyze the forces acting on the rod connected to the pivot. The tension in this rod must balance the centripetal force of both masses. The inner mass contributes a force based on its distance from the pivot, and the outer mass contributes a force based on its distance from the pivot.

Step 4: Add the contributions of the centripetal forces from both masses to find the total tension in the rod connected to the pivot. Use the formula $ T = F_{c, \text{inner}} + F_{c, \text{outer}} $.

Step 5: Substitute the values for mass, radius, and angular velocity into the equations to calculate the tension. Ensure the radii are converted to meters (e.g., 75 cm = 0.75 m) before performing calculations.

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Key Concepts

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

Centripetal Force

Centripetal force is the net force that acts on an object moving in a circular path, directed towards the center of the circle. In this scenario, the tension in the rod provides the necessary centripetal force to keep the masses rotating. The formula for centripetal force is F_c = m * v^2 / r, where m is the mass, v is the tangential velocity, and r is the radius of the circular path.

Rotational Motion

Rotational motion refers to the motion of an object that rotates around an axis. In this case, the two masses are rotating around the rotor at a specified angular velocity (30 rpm). Understanding rotational motion involves concepts such as angular velocity, which is the rate of change of angular displacement, and can be converted to linear velocity using the relationship v = ω * r, where ω is the angular velocity in radians per second.

Tension in a Rod

Tension in a rod is the force exerted along the length of the rod when it is subjected to forces, such as those from rotating masses. In this problem, the tension in the rod connecting the masses to the pivot must counteract the outward force due to the circular motion of the masses. The tension can be calculated by analyzing the forces acting on one of the masses and applying Newton's second law in the context of circular motion.

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Gravitational Force of Rod Parallel to Axis

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

Suppose you swing a ball of mass m in a vertical circle on a string of length L. As you probably know from experience, there is a minimum angular velocity ω_min you must maintain if you want the ball to complete the full circle without the string going slack at the top. Find an expression for ω_min.