Question

2. [1pt] Answer True, False, or Cannot tell to each of the four statements below. E.g., if the answer to the first statement is 'true' and to the rest, cannot tell', enter TCCC. a. The solid cylinders and the cylindrical shells have the same mass, the same radius, and turn on frictionless axles. A rope is wrapped around each of them and is acted on as shown. Cylinder shell 3 has the smallest angular acceleration. b. Force F and force F, exert the same torque about the axle. (Note: F = F) C. A square plate can rotate about an axle through its centre. One of three forces of equal magnitude can be applied to the plate, as shown. The angular acceleration of the plate is the same whichever of the forces is applied. d. A wheel turns on a frictionless axle. A string wrapped around the rim of the wheel is connected to a mass. When the mass is released, the magnitude of its acceleration is less than g.

Answer 1

a)

For the four ( 2 solid cylinders, 2 hollow cylinders) same amount of force 10 N is acting downwards.

For hanging mass, weight is  

mg = 10 m=

For 1st solid cylinder:

Net force acting on hanging mass is
10 10-T= -a 9
and torque of the solid cylinder about its center is
T= lg = TR
, $\alpha=\frac{a}{R}$ ,
MR I =
( M is mass of solid cylinder, R is radius of solid cylinder)

Using above relations,  

1

For the 2nd solid cylinder:

Net force acting string is
T = 10
and torque of the solid cylinder about its center is
T= lg = TR
, $\alpha=\frac{a}{R}$ ,
MR I =
( M is mass of solid cylinder, R is radius of solid cylinder)

Using above relations,  

= D

For 3rd hollow cylinder:

Net force acting on hanging mass is
10 10-T= -a 9
and torque of the cylindrical shell  about its center is
T= lg = TR
, $\alpha=\frac{a}{R}$ ,
I = MR
( M is mass of cylindrical shell , R is radius of cylindrical shell)

Using above relations,  

10 = 10 + M

For 4th hollow cylinder:

Net force acting on hanging mass is
T = 10
and torque of the cylindrical shell  about its center is
T= lg = TR
, $\alpha=\frac{a}{R}$ ,
I = MR
( M is mass of cylindrical shell , R is radius of cylindrical shell)

Using above relations,  $a=\frac{10 }{M}$

From above results acceleration is smallest for 3rd cylindrical shell.

Angular acceleration is $\alpha=\frac{a}{R}$ , so angular acceleration is smallest for cylindrical shell 3.

Answer is true.

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b,

Force $F_1$ acts at a distance of $r_1$ from the center and force $F_2$ acts at a distance of $r_2$ from the center.

Torque exerted by F₁ about the axle is

T1 = r11

Torque exerted by F₂ is   ( since F₁ = F₂ )

T2 = r2 F2sin 30º =

As the distances  $r_1,r_2$ are not known, one cannot say whether the two torques are equal or not.

Answer is cannot tell.

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c.

F1 = F2 = F3

$r_2$ is the perpendicular distance of force $F_2$ from center of plate.

$r_3$ is the perpendicular distance of force $F_3$ from center of plate.

from the figure $r_3>r_2$

The force F₃ passes through the center of square plate , torque due to force F₁ is zero.

T1 = 0

Torque due to force F₂ is

To = r2F

Torque due to force F₃ is   

T3 = 13 F3

Т3 > Т2 > T1

Angular acceleration of plate is $\alpha =\frac{\tau}{I}$

=0
, $\alpha _2=\frac{\tau_2}{I}=\frac{r_2F_2}{I}$ , $\alpha _3=\frac{\tau_3}{I}=\frac{r_3F_3}{I}$

$r_3>r_2$

a3 > 02 > 01

Answer is False.

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d)

Net force acting on hanging mass is
mg-T=ma
and torque of the wheel about its center is
T= lg = TR
, $\alpha=\frac{a}{R}$ ,   ( R is radius of wheel)

using the above relations,

a=_mg - m+ I <g

Answer is True.

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Final answer:

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TCFT