Ch 12: Rotation of a Rigid Body

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

Ch 12: Rotation of a Rigid Body

Problema 27

Capitolo 12, Problema 27

A 12-cm-diameter, 600 g cylinder, initially at rest, rotates on an axle along its axis. A steady 0.50 N force applied tangent to the edge of the cylinder causes the cylinder to reach an angular velocity of 500 rpm in 2.0 s. What is the magnitude of the frictional torque between the cylinder and the axle?

Guida verificata passo dopo passo

Determine the moment of inertia of the cylinder. For a solid cylinder rotating about its central axis, the moment of inertia is given by $ I = \frac{1}{2} m r^2 $, where $ m $ is the mass of the cylinder and $ r $ is its radius. Convert the diameter (12 cm) to radius (6 cm or 0.06 m) and use the mass (600 g or 0.6 kg) to calculate $ I $.

Calculate the angular acceleration $ \alpha $ of the cylinder. Use the kinematic equation $ \omega_f = \omega_i + \alpha t $, where $ \omega_f $ is the final angular velocity, $ \omega_i $ is the initial angular velocity (0 rad/s since it starts from rest), and $ t $ is the time (2.0 s). Convert the final angular velocity from rpm (500 rpm) to rad/s using $ \omega_f = \text{rpm} \times \frac{2\pi}{60} $. Solve for $ \alpha $.

Determine the net torque acting on the cylinder using the rotational analog of Newton's second law: $ \tau_{\text{net}} = I \alpha $. Substitute the values of $ I $ and $ \alpha $ calculated in the previous steps to find $ \tau_{\text{net}} $.

Calculate the applied torque due to the tangential force. Torque is given by $ \tau_{\text{applied}} = F r $, where $ F $ is the tangential force (0.50 N) and $ r $ is the radius of the cylinder (0.06 m).

Find the frictional torque by subtracting the applied torque from the net torque. Use $ \tau_{\text{friction}} = \tau_{\text{applied}} - \tau_{\text{net}} $. The result will give the magnitude of the frictional torque opposing the motion.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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 scenario, the torque generated by the applied force can be determined using the formula τ = r × F, where τ is torque, r is the radius of the cylinder, and F is the force applied tangentially.

Video consigliato:

Percorso guidato

08:55

Net Torque & Sign of Torque

Moment of Inertia

The moment of inertia is a property of a body that quantifies its resistance to angular acceleration about an axis. For a solid cylinder, it is calculated using the formula I = (1/2) m r², where m is the mass and r is the radius. Understanding the moment of inertia is crucial for analyzing how the cylinder responds to applied torques.

Video consigliato:

Percorso guidato

11:47

Intro to Moment of Inertia

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). It can be calculated using the formula α = (ω_f - ω_i) / t, where ω_f is the final angular velocity, ω_i is the initial angular velocity, and t is the time taken. This concept is essential for determining the net torque acting on the cylinder, including the effects of friction.

Video consigliato:

Percorso guidato

12:12

Conservation of Angular Momentum

Pratica correlata

Domanda del libro di testo

A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid, massless rod. The rod is rotating cw about its center of mass at 20 rpm. What net torque will bring the balls to a halt in 5.0 s?

4425

views

Domanda del libro di testo

The object shown in FIGURE EX12.29 is in equilibrium. What are the magnitudes of $F sub 1 right arrow sub$ and $F sub 2 right arrow sub$ ?

3217

views

Domanda del libro di testo

A 5.0 kg cat and a 2.0 kg bowl of tuna fish are at opposite ends of the 4.0-m-long seesaw of FIGURE EX12.32. How far to the left of the pivot must a 4.0 kg cat stand to keep the seesaw balanced?

2211

views

Domanda del libro di testo

An object whose moment of inertia is 4.0 kg m² is rotating with angular velocity 0.25 rad/s. It then experiences the torque shown in FIGURE EX12.25. What is the object's angular velocity at t = 3.0s?

3322

views

rank

Domanda del libro di testo

An object's moment of inertia is 2.0 kg m². Its angular velocity is increasing at the rate of 4.0 rad/s per second. What is the net torque on the object?

2251

views

Domanda del libro di testo

A car tire is 60 cm in diameter. The car is traveling at a speed of 20 m/s. What is the speed of a point at the top edge of the tire?

2353

views