**Problem: Rolling Motion Analysis****Description:**A disk with radius \( R \) and mass \( m \) begins from rest and then moves without slipping while being pulled horizontally by a force \( P \) acting at its center axle. Demonstrate that the velocity of the wheel after \( T \) seconds is \( v = \frac{2PT}{3m} \). *(Hint: use both linear and angular-impulse principles.)***Diagram Explanation:**The diagram shows a disk rolling on a horizontal surface. The center of the disk is labeled as \( G \) with a force \( P \) acting horizontally through this point. An arrow indicates the direction of the force. The radius \( R \) extends from the center to the edge of the disk.**Additional Information on Radius of Gyration:**The radius of gyration has units of length and is related to the inertia by \( k_G^2 = \frac{I_G}{m} \). It represents the distance at which a mass equivalent to that of the rigid body would produce the same inertia as the rigid body itself. Recall that the inertia of a particle of mass \( m \) at a distance \( r \) from an axis of rotation is \( mr^2 \). Rather than using \( r \), the convention is to define the radius of gyration with the symbol \( k \).

In the above diagram, N is normal force, m is mass, g is gravitational acceleration, P is applied…

A disk with radius R and mass m begins from rest and then moves without slipping while being pulled horizontall by a force P acting at its center axle. Show that the velocity of the wheel after T seconds is v = 2PT/3m. (Hint: use both linear and angular-impulse principles.) m R P ¹The radius of gyration has units of length and is related to the inertia by k = Ic/m. It corresponds to the distance at which a mass equivalent to the mass of the rigid body would produce the same inertia as the actual rigid body. Recall that the inertia of a particle of mass m at a distance r from an axis of ortation is mr². Rather that using r the convention is to define the radus of gyration with the symbol k.

A disk with radius R and mass m begins from rest and then moves without slipping while being pulled horizontall by a force P acting at its center axle. Show that the velocity of the wheel after T seconds is v = 2PT/3m. (Hint: use both linear and angular-impulse principles.) m R P ¹The radius of gyration has units of length and is related to the inertia by k = Ic/m. It corresponds to the distance at which a mass equivalent to the mass of the rigid body would produce the same inertia as the actual rigid body. Recall that the inertia of a particle of mass m at a distance r from an axis of ortation is mr². Rather that using r the convention is to define the radus of gyration with the symbol k.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Combined Loading

Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial forces, torsion, and so on, then such body is said to be under combined loading. A very practical example of combined loading is the crankshaft of an internal combustion engine (IC engine). The crankshaft of the IC engine is under high rpm, which is caused due to the torsional forces. Due to the presence of piston, counterbalances, and internal imbalances, the crankshaft experiences lateral forces and due to the rise in temperature, the crankshaft undergoes thermal expansion, which pulls the crankshaft along the axial direction and lateral direction. The presence of unbalanced loads along the periphery of the crankshaft also induces a bending moment.

Question

**Problem: Rolling Motion Analysis**

**Description:**

A disk with radius \( R \) and mass \( m \) begins from rest and then moves without slipping while being pulled horizontally by a force \( P \) acting at its center axle. Demonstrate that the velocity of the wheel after \( T \) seconds is \( v = \frac{2PT}{3m} \).

*(Hint: use both linear and angular-impulse principles.)*

**Diagram Explanation:**

The diagram shows a disk rolling on a horizontal surface. The center of the disk is labeled as \( G \) with a force \( P \) acting horizontally through this point. An arrow indicates the direction of the force. The radius \( R \) extends from the center to the edge of the disk.

**Additional Information on Radius of Gyration:**

The radius of gyration has units of length and is related to the inertia by \( k_G^2 = \frac{I_G}{m} \). It represents the distance at which a mass equivalent to that of the rigid body would produce the same inertia as the rigid body itself. Recall that the inertia of a particle of mass \( m \) at a distance \( r \) from an axis of rotation is \( mr^2 \). Rather than using \( r \), the convention is to define the radius of gyration with the symbol \( k \).

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