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Chapter 9: Q26P (page 378)

Show that moment of inertia of a disk of mass M and radius R is $\frac{1}{2} {MR}^{2}$ . Divide the disk into narrow rings, each of radius r and width dr. The contribution I of by one of these rings is r²dm, where dm is amount of mass contained in that particular ring. The mass of any ring is the total mass times the fraction of the total area occupied by the area of the ring. The area of this ring is approximately $2 蟺谤诲谤$ . Use integral calculus to add up all the calculations.

Short Answer

Expert verified

It is proved that the moment of inertia of disk is $\frac{1}{2} M R^{2}$ .

Step by step solution

01

Identification of given data

The mass of disk is $M$ .

The radius of disk is $R$ .

The radius of each ring is $r$ .

The width of each ring is $d r$ .

The mass of each ring is $d m$ .

The moment of inertia of each ring is $I = r^{2} d m$ .

02

Conceptual Explanation

The moment of inertia of the disk is obtained by calculating mass of each ring then substitute in the formula for mass of each ring.

03

Determination of moment of inertia of disk

The density of disk is given as:

$蚁 = \frac{M}{(蟺 R^{2}) r}$

The volume of each ring is given as:

$V = (2 蟺 r d r) r d V = 2 蟺 r^{2} d r$

The mass of each ring is given as:

$d m = 蚁 d V d m = (\frac{M}{(蟺 R^{2}) r}) (2 蟺 r^{2} d r) d m = (\frac{2 M}{R^{2}}) (r d r)$

The moment of inertia of disk is calculated as:

$I_{d} = {鈭�}_{0}^{R} I I_{d} = {鈭�}_{0}^{R} r^{2} d m I_{d} = {鈭�}_{0}^{R} r^{2} ((\frac{2 M}{R^{2}}) (r d r)) I_{d} = (\frac{2 M}{R^{2}}) {鈭�}_{0}^{R} r^{3} d r$

$I_{d} = (\frac{2 M}{R^{2}}) {(\frac{r^{4}}{4})}_{0}^{R} I_{d} = (\frac{2 M}{R^{2}}) (\frac{R^{4}}{4}) I_{d} = \frac{1}{2} M R^{2}$

Therefore, the moment of inertia of disk is $\frac{1}{2} M R^{2}$ .

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Most popular questions from this chapter

A runner whose mass is $50 kg$ accelerates from a stop to a speed of $10 m / s$ in $3 s$ . (A good sprinter can run $100 m$ in about $10 s$ , with an average speed of $10 m / s$ .) (a) What is the average horizontal component of the force that the ground exerts on the runner鈥檚 shoes? (b) How much displacement is there of the force that acts on the sole of the runner鈥檚 shoes, assuming that there is no slipping? Therefore, how much work is done on the extended system (the runner) by the force you calculated in the previous exercise? How much work is done on the point particle system by this force? (c) The kinetic energy of the runner increases鈥攚hat kind of energy decreases? By how much?

A solid uniform-density sphere is tied to a rope and moves in a circle with speed $v$ . The distance from the center of the circle to the center of the sphere is $d$ , the mass of the sphere is $M$ , and the radius of the sphere is $R$ . (a) What is the angular speed $蝇$ ? (b) What is the rotational kinetic energy of the sphere? (c) What is the total kinetic energy of the sphere?

Two disks are initially at rest, each of mass M, connected by a string between their centers, as shown in Figure 9.55. The disks slide on low-friction ice as the center of the string is pulled by a string with a constant force F through a distance d. The disks collide and stick together, having moved a distance b horizontally.

(a) What is the final speed of the stuck-together disks? (b) When the disks collide and stick together, their temperature rises Calculate the increase in internal energy of the disks assuming that the process is so fast that there is insufficient time for there to be much transfer of energy to the ice due to a temperature difference. (Also ignore the small amount of energy radiated away as sound produced in the collisions between the disks)

Three uniform-density spheres are positioned as follows:

A $3 k g$ sphere is centered at $< 10, 20, - 5 > m$ .
A $5 k g$ sphere is centered at $< 4, - 15, 8 > m$ .
A $6 k g$ sphere is centered at $< - 7, 10, 9 > m$ .

What is the location of the center of mass of this three-sphere system?

A uniform-density disk whose mass is 10 kg and radius is 0.4 m makes one complete rotation every 0.2 s. What is the rotational kinetic energy of the disk?

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