Principles of Mechanics pp 103-122 | Cite as

# Rotation of Rigid Bodies

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## Abstract

Rotational motion exists everywhere in the universe. The motion of electrons about an atom and the motion of the moon about the earth are examples of rotational motion. Objects cannot be treated as particles when exhibiting rotational motion since different parts of the object move with different velocities and accelerations. Therefore, it is necessary to treat the object as a system of particles.

## 7.1 Rotational Motion

Rotational motion exists everywhere in the universe. The motion of electrons about an atom and the motion of the moon about the earth are examples of rotational motion. Objects cannot be treated as particles when exhibiting rotational motion since different parts of the object move with different velocities and accelerations. Therefore, it is necessary to treat the object as a system of particles.

## 7.2 The Plane Motion of a Rigid Body

- 1.
The pure rotational motion: The rigid body in such a motion rotates about a fixed axis that is perpendicular to a fixed plane. In other words, the axis is fixed and does not move or change its direction relative to an inertial frame of reference.

- 2.
The general plane motion: The motion here can be considered as a combination of pure translational motion parallel to a fixed plane in addition to a pure rotational motion about an axis that is perpendicular to that plane. This chapter discusses the kinematics and dynamics of pure rotational motion.

### 7.2.1 The Rotational Variables

*r*which represents the perpendicular distance from \(\mathrm {P}\) to the axis of rotation. If you look at any other particle in the object you will see that every particle will rotate in its own circle that has the axis of rotation at its center. In other words, different particles move in different circles but the center of all of these circles lies on the rotational axis. Suppose the particle moves through an arc length

*s*starting at the positive \(\mathrm {x}\)-axis. Its angular position is then given by

*r*and \(\theta \) are the polar coordinates of a point in a plane (which was mentioned in Sect. 2.6) where \(\theta \) is always measured from the positive \(\mathrm {x}\)-axis. Because \(\theta \) is the ratio of the arc length to the radius, it is a pure (dimensionless) number. The unit usually used to measure \(\theta \) is the radians (rad). One radian is defined as the angle subtended by an arc of length that is equal to the radius of the circle. Since one rotation (\(360^{\circ }\)) corresponds to \(\theta =2\pi r/r=2\pi \) rad, it follows that:

*x*,

*v*and

*a*in translational one-dimensional motion. The vectors \(\omega \) and \(\alpha \) are not used in the case of pure rotational motion, they are used in the general rotational motion when the axis of rotation changes its direction with time. Note that only the infinitesimal angular displacement \( d\theta \) can be represented by a vector but not the finite angular displacement \(\triangle \theta \). This is because the finite angular displacement \(\triangle \theta \) does not obey the commutative law of vector addition (see Fig. 7.5) and therefore cannot be represented by a vector. Hence, the instantaneous angular velocity and acceleration (\(\omega \) and \(\alpha \)) can be represented by vectors but not their average values (\(\overline{\omega }\) and \(\overline{\alpha }\)).

### Example 7.1

Convert each of the following into the other angular units: \(15^\circ \), 0.25 \(\mathrm {r}\mathrm {e}\mathrm {v}/\mathrm {s}^{2}\), 3 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}.\)

### Solution 7.1

### Example 7.2

A rotating rigid object has an angular position given by \(\theta (t)=((0.3)t^{2}+(0.4)t^{3})\) rad. Determine: (a) the angular displacement of the object and the average angular velocity during the time interval from \(t_{1}=1\mathrm {s}\) to \(t_{2}=2 \; \mathrm {s} \). (b) the instantaneous angular velocity and the instantaneous angular acceleration at \(t=5 \; \mathrm {s}\).

### Solution 7.2

### Example 7.3

A wheel is rotating with an angular acceleration that is given by \(\alpha =(9-2t) \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}^{2}\). (a) Find the angular velocity and displacement at any time if at \(t=0\) the wheel has an angular velocity of 2 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\) and an (initial) angular displacement of 3 rad; (b) at what angular displacement will the wheel reach its maximum angular velocity

### Solution 7.3

## 7.3 Rotational Motion with Constant Acceleration

*t*from Eq. 7.1 and substituting into Eq. 7.2 gives

*s*along its circular path (see Fig. 7.6). The angular displacement of the particle is related to

*s*by

*r*is the radius of the circle in which the particle is moving along. Differentiating the above equation with respect to

*t*gives

*ds*/

*dt*is the magnitude of the linear velocity of the particle and \(d\theta /dt\) is the angular velocity of the body we may write

Kinematic equations

Rotational motion about a fixed axis with constant \(\alpha \) | Linear motion with constant |
---|---|

\(\omega =\omega _{0}+\alpha t\) | \(v=v_{0}+at\) |

\(\displaystyle \theta =\theta _{0}+\frac{1}{2}(\omega +\omega _{0})t^{} \) | \(x=x_{0}+\displaystyle \frac{1}{2}(v+v)t_{} \) |

\(\displaystyle \theta =\theta _{0}^{}+\omega _{0}t+\frac{1}{2}\alpha t^{2}\) | \(x=x_{0}+v_{0}t_{}+\displaystyle \frac{1}{2}at^{2}\) |

\( \omega ^{2}=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\) | \( v^{2}=v_{0}^{2}+2a(x-x_{0})\) |

### Example 7.4

A disc of radius of 10 cm rotates from rest with a constant angular acceleration. If it requires 2 \(\mathrm {s}\) for it to rotate through an angular displacement of \(60^{\mathrm {o}}\): (a) find the angular acceleration of the disc; (b) its angular velocity at \(t=2\mathrm {s}\) and at \(t=6\mathrm {s}, (\mathrm {c})\) the linear speed at \(t=2\mathrm {s}\) of a point that is at a distance of 7 cm from the center of the disc; (d) the distance that this point has moved during that time interval.

### Solution 7.4

### Example 7.5

Two sprockets are attached to each other as in Fig. 7.8. There radii are \(r_{1}= 2\) cm and \(r_{2}=5\) cm. If the angular velocity of the smaller sprocket is 2 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s},\) find the angular velocity of the other.

### Solution 7.5

### Example 7.6

Find the angular speed of the moon in its orbit about the earth in rev/day.

### Solution 7.6

*r*is the mean distance from the earth to the moon and

*T*is its period. Thus, the angular velocity of the moon is

## 7.4 Vector Relationship Between Angular and Linear Variables

## 7.5 Rotational Energy

*i*th particle respectively (see Fig. 7.10). From Eq. 7.5, we have

*I*changes as well. The SI unit of the moment of inertia is kg \(\mathrm {m}^{2}\). The rotational kinetic energy can thus be written as

*dm*in terms of its position coordinates.

## 7.6 The Parallel-Axis Theorem

*I*of a system about any axis that is parallel to an axis passing through the center of mass is

*M*is the total mass of the system, and

*D*is the perpendicular distance between the two parallel axes.

### Proof

*x*and

*y*are the coordinates of the mass element

*dm*from the center of mass (the origin). Now consider another axis that is parallel to the first axis and that passes through a point \(\mathrm {P}\) as shown in Fig. 7.11. Suppose that the \(\mathrm {x}\) and \(\mathrm {y}\) coordinates of \(\mathrm {P}\) from the center of mass are \(x_{p}\) and \(y_{p}\). The moment of inertia about an axis passing through \(\mathrm {P}\) is

*dm*from point P Expanding this equation gives

**Special Moment of Inertia**Fig. 7.12 gives the rotational inertia of various rigid bodies of uniform density.

## 7.7 Angular Momentum of a Rigid Body Rotating about a Fixed Axis

*I*is the moment of inertia of the rigid body about the rotational axis (z-axis). This equation can also be written in component form since \(\mathbf {L}_{z}\) is parallel to \(\varvec{\omega }\), that is,

### Example 7.7

A 5 kg wheel of radius of 0.1 \(\mathrm {m}\) decelerates from an angular speed of 5 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\) to rest after going through an angular displacement of 10 rev If a frictional force causes the wheel to decelerate, find the torque due to this force.

### Solution 7.7

### Example 7.8

Three masses are connected by massless rods as in Fig. 7.15. If \(m=0.1 \; \mathrm {k}\mathrm {g},\) find the moment of inertia of the system and the corresponding kinetic energy if it rotates with an angular speed of 5 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\) about: (a) the \(\mathrm {z}\)-axis; (b) the \(\mathrm {y}\)-axis and; (c) the \(\mathrm {x}\)-axis \((a=0.2 \; \mathrm {m})\).

### Solution 7.8

### Example 7.9

Fig. 7.16 shows a uniform thin rod of mass *M* and length *L*. Find the moment of inertia of the rod about an axis that is perpendicular to it and passing through: (a) the center of mass; (b) at one end; (c) at a distance of *L* / 6 from one end.

### Solution 7.9

*dm*of an element in the rod is

### Example 7.10

Fig. 7.17 shows a uniform thin plate of mass *M* and surface density \(\sigma \). Find the moment of inertia of the plate about an axis passing through its center of mass if its length is *b* and its width is *a* (the \(\mathrm {z}\)-axis).

### Solution 7.10

*dm*has an area

*dxdy*and is at a distance \(r=\sqrt{x^{2}+y^{2}}\) from the axis of rotation. Therefore, we have

### Example 7.11

Find the moment of inertia of a uniform solid cylinder of radius *R*, length *L* and mass *M* about its axis of symmetry.

### Solution 7.11

*r*, length

*L*and thickness

*dr*as in Fig. 7.18, then each volume element is given by

### Example 7.12

Three rods of length *L* and mass *M* are connected together as in Fig. 7.19. Determine the moment of inertia of the system about an axis passing through \(\mathrm {O}\) and perpendicular to the page (the rods lie in the same plane).

### Solution 7.12

### Example 7.13

Find the moment of inertia of a spherical shell of radius *R* and mass *M* about an axis passing through its center of mass.

### Solution 7.13

## 7.8 Conservation of Angular Momentum of a Rigid Body Rotating About a Fixed Axis

## 7.9 Work and Rotational Energy

**The Work–Energy Theorem**The work–energy theorem states that the work done by an external force while a rigid object rotate from \(\theta _{1}\) to \(\theta _{2}\) is equal to the change in the rotational energy of the object. This follows from Eq. 7.12 and by using the fact that along the axis of rotation the torque is given by \(\tau _{z}=I\alpha \) (see Sect. 7.7), thus

Analogous Equations in linear Motion and Rotational Motion about a Fixed Axis

Rotational motion | Linear motion |
---|---|

\(\tau =I\alpha \) | \(F=ma\) |

\(W=\int _{\theta _{0}}^{\theta }\tau d\theta \ \) | \( W=\int _{x_{0}}^{x}Fdx\) |

\(K_{R}=\frac{1}{2}I\omega ^{2}\) | \(K=\frac{1}{2}mv^{2}\) |

\( P=\tau \omega \) | \(P=Fv\) |

## 7.10 Power

### Example 7.14

A disc of radius \(R=0.08 \; \mathrm {m}\) and mass of 5 kg is rotating about its central axis with an angular speed of 170 rev/min. Find: (a) the rotational kinetic energy of the disc; (b) Suppose that the same disc rotate using a motor that delivers an instantaneous of power 0. \(2\mathrm {h}\mathrm {p}\), find in that case the torque applied to the disc.

### Solution 7.14

### Example 7.15

Consider a light rope wrapped around a uniform cylindrical shell of mass 30 kg and radius of 0.2 \(\mathrm {m}\) as in Fig. 7.22. Suppose that the cylinder is free to rotate about its central axis and that the rope is pulled from rest with a constant force of magnitude of 35 N. Assuming that the rope does not slip, find: (a) the torque applied to the cylinder about its central axis; (b) the angular acceleration of the cylinder; (c) the acceleration of a point in the unwinding rope; (d) the number of revolutions made by the cylinder when it reaches an angular velocity of 12 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}, (\mathrm {e})\) the work done by the applied force when the rope is pulled a distance of \(1\mathrm {m}, (\mathrm {f})\) the work done using the work–energy theorem.

### Solution 7.15

### Example 7.16

A uniform rod of mass \(M=0.75\) kg and length \(L=1\mathrm {m}\) is hinged at one end and is free to rotate in a vertical plane as in Fig. 7.23. If the rod is released from rest at an angle \(\theta =30^{\mathrm {o}}\) to the horizontal, find; (a) the initial angular acceleration of the rod when it is released; (b) the initial acceleration of a point at the end of the rod; (c) from conservation of energy find the angular speed of the rod at its lowest position (Neglect friction at the pivot).

### Solution 7.16

### Example 7.17

Find the net torque on the system shown in Fig. 7.24 where \(r_{1}=5\) cm, \(r_{2}=15\) cm, \(F_{1}=10 \; \mathrm {N}, F_{2}=20 \; \mathrm {N}\) and \(F_{3}=15 \; \mathrm {N}\). Neglect the mass and friction of the ropes and pulleys.

### Solution 7.17

### Example 7.18

A block of mass *m* is attached to a light string that is wrapped around the rim of a uniform solid disc of radius *R* and mass *M* as in Fig. 7.25. Assuming that the string does not slip and that the disc rotates without friction, find: (a) the acceleration of the block; (b) the angular acceleration of the disc, and; (c) the tension in the string when the system is released from rest.

### Solution 7.18

### Example 7.19

A homogeneous solid sphere of mass 4.7 kg and radius of 0.05 \(\mathrm {m}\) rotate from rest about its central axis with a constant angular acceleration of 3 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}^{2}\). Find: (a) the torque that produces this angular acceleration; (b) the work done on the sphere after 7 revolutions; (c) the work done after 7 revolutions using the work–energy theorem.

### Solution 7.19

### Example 7.20

Fig. 7.26 shows Atwood’s machine when the mass of the pulley is considered. If the system is released from rest (and assuming that the string does not stretch or slip) and that the friction of the pulley is negligible, find linear acceleration of the blocks and the angular acceleration of the pulley.

### Solution 7.20

### Example 7.21

A uniform solid cylinder of radius of 0.2 \(\mathrm {m}\) and mass of 10 kg is rotating about its central axis. If the angular speed of the cylinder is 5 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}{:} (\mathrm {a})\) calculate the angular momentum of the cylinder about its central axis; (b) Suppose the cylinder accelerates at a constant rate of 0.5 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}^{2}\), find the angular momentum of the cylinder at \(t=3\mathrm {s}(\mathrm {c})\) find the applied torque; (d) find the work done after \(3\mathrm {s}.\)

### Solution 7.21

### Example 7.22

A uniform solid sphere of radius of 5 cm and mass of 4.7 kg is rotating about an axis that is tangent to the sphere (see Fig. 7.27). If its angular acceleration is given by \(\alpha =(4t)\,\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}^{2}\) and if at \(t=0, \omega _{0}=0\), find the angular momentum of the sphere and the applied torque as a function of time.

### Solution 7.22

### Example 7.23

In Example 7.8 find the angular momentum in each case.

### Solution 7.23

### Example 7.24

A uniform solid sphere of radius of 0.2 \(\mathrm {m}\) is rotating about its central axis with an angular speed of 5 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\). If an impulsive force that has an average value of 100 \(\mathrm {N}\) acts at the rim of the sphere at the center level for a short time of 2 \(\mathrm {m}\mathrm {s}\):\((\mathrm {a})\) find the angular impulse of the force; (b) the final angular speed of the sphere.

### Solution 7.24

(a)

\(\displaystyle \triangle L=\int _{t_{1}}^{t_{2}}\tau dt=\tau _{ave}\triangle t=\overline{F}Rt=(100 \; \mathrm {N})(0.2 \; \mathrm {m})(2\times 10^{-3} \; \mathrm {s})=0.04 \; \mathrm {k}\mathrm {g}\,\mathrm {m}^{2}/\mathrm {s}\)

### Example 7.25

A man stands on a platform that is free to rotate without friction about a vertical axis as in Fig. 7.28. If the system is initially rotating with an angular speed of 0.3 \(\mathrm {r}\mathrm {e}\mathrm {v}/\mathrm {s}{:}\,(\mathrm {a})\) find the final angular speed of the system if the man draws the weights in; (b) find the increase in the kinetic energy of the system and its source. \((I_{i}=15 \; \mathrm {kg\, m^2}\) And \(I_{f}=3 \; \mathrm {k}\mathrm {g}\,\mathrm {m}^{2})\).

### Solution 7.25

### Example 7.26

A uniform disc of moment of inertia of 0.1 kg m\(^{2}\) is rotating without friction with an angular speed of 3 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\) about an axle passing through its center of mass as in Fig. 7.29. When another disc of moment of inertia of 0.05 kg m\(^{2}\) that is initially at rest is dropped on the first, the two will eventually rotate with the same angular speed due to friction between them. Determine (a) the final angular speed; (b) the change in the kinetic energy of the system.

### Solution 7.26

**Problems**

- 1.
A wheel is initially rotating at 60 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\) in the clockwise direction. If a counterclockwise torque acts on the wheel producing a counterclockwise angular acceleration \(\alpha =2t \; \mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}^{2}\), find the time required for the wheel to reverse its direction of motion.

- 2.
If the angular position of a point on a rotating wheel is given by \(\theta =2t+ 5t^{2}\) rad, find the angular speed and angular acceleration of the point at \(t=2 \; \mathrm {s}.\)

- 3.
A wheel of radius of 0.5 \(\mathrm {m}\) rotates from rest at a constant angular acceleration of 2.5 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}^{2}\). At \(t=2 \; \mathrm {s}\) Find (a) the angular speed of the wheel (b) the angle in radians through which the wheel rotates (c) the tangential and radial acceleration of a point at the rim of the wheel.

- 4.
Find the angular speed in radians per second of the earth about (a) its axis (b) the sun.

- 5.
An \(\mathrm {L}\)-shaped bar rotates counterclockwise with an angular acceleration of \(\omega \) (see Fig. 7.30). Find (in vector form) the linear velocity and acceleration of the point \(\mathrm {P}\) on the bar.

- 6.
Four masses are connected by light rigid rods as in Fig. 7.31. Calculate the moment of inertia of the system about (a) the \(\mathrm {x}\)-axis (b) the \(\mathrm {y}\)-axis (c) the \(\mathrm {z}\)-axis.

- 7.
Find the moment of inertia of a uniform solid sphere of radius

*R*and mass*M*about an axis passing through its center of mass. - 8.
Find the moment of inertia of an elliptical quadrant about the \(\mathrm {y}\)-axis (see Fig. 7.32).

- 9.
A 5 kg uniform solid cylinder of radius 0.2 \(\mathrm {m}\) rotate about its center of mass axis with an angular speed of 10 rev/min. Find (a) its rotational kinetic energy (b) its angular momentum.

- 10.
A wheel of mass of 20 kg and radius of 0.75 \(\mathrm {m}\) is initially rotating at 120 rev/min. If its angular speed is increased to 300 rev/min in 20 \(\mathrm {s}\), find (a) the work done on the wheel (b) the average power delivered to the wheel.

- 11.
A wheel of mass 10 kg and radius 0.4 \(\mathrm {m}\) accelerates uniformly from rest to an angular speed of 800 rev/min in 20 \(\mathrm {s}\). Find (a) the torque applied to the wheel (b) the work done on the wheel (c) the work done using the work–energy theorem.

- 12.
A uniform rod of length

*L*and mass*M*is pivoted at \(\mathrm {O}\) (see Fig. 7.33). If a projectile of mass*m*moving at velocity*v*collide with the rod and stick to it, find the angular momentum of the system immediately before and immediately after the collision. - 13.
A disc of radius 2.2 \(\mathrm {m}\) and mass of 120 kg rotate about a frictionless vertical axle that passes through its center. A man of mass 65 kg walks slowly from the rim of the disc towards the center. Find the angular speed of the disc when the man is at a distance of 0.7 \(\mathrm {m}\) from the center if its angular speed when the man starts walking is 1.6 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}.\)

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