Question

You have a pulley 10.4 cm in diameter and with a mass of 2.3 kg. You get to wondering whether the pulley is uniform. That is, is the mass evenly distributed, or is it concentrated toward the center or the rim? To find out, you hang the pulley on a hook, wrap a string around it several times, and suspend your 1.7 kg physics book 1.0 m above the floor. With your stopwatch, you can find that it takes 0.64 s for your book to hit the floor. Use g=9.8 m/s2.

Answer 1

Answer:

Explanation:

Given

Diameter of Pulley=10.4 cm

mass of Pulley(m)=2.3 kg

mass of book$(m_0)=1.7 kg$

height(h)=1 m

time taken=0.64 s

$h=ut+frac{at^2}{2}$

$1=0+\frac{a(0.64)^2}{2}$

$a=4.88 m/s^2

and [tex]a=\alpha r$

where $\alpha$is angular acceleration of pulley

$4.88=\alpha \times 5.2\times 10^{-2}$

$\alpha =93.84 rad/s^2$

And Tension in Rope

$T=m(g-a)$

$T=1.7\times (9.8-4.88)$

T=8.364 N

and Tension will provide Torque

$T\times r=I\cdot \alpha$

$8.364\times 5.2\times 10^{-2}=I\times 93.84$

$I=0.463\times 10^{-2} kg-m^2$

$I_{original}=\frac{mr^2}{2}=0.31\times 10^{-2}kg-m^2$

Thus mass is uniformly distributed or some more towards periphery of Pulley

Answer 2

The kinematics and Newton's second law for linear and rotational motions allow us to find the result for the mass distribution is:

• The mass is distributed more towards the edge of the pulley.

Kinematics studies the movement of bodies, finding relationships between the position, speed and acceleration of bodies.

Newton's second law for rotational motion says that torque is directly proportional to the moment of inertia times the angular acceleration.

Let's solve this problem in parts.

1st part. Let's look for the acceleration of the body when it falls to the ground.

Indicate that the height of the body is i = 1.0 m and the fall time is t = 0.64 s, let's use the relation

            y = y₀ + v₀ t - ½ a t²

Where y and y₀ are the current and initial height, respectively, v₀ the initial velocity, t the time and a  the fall acceleration that due to the pulley can be different from the gravity acceleration.

As it comes of the rest, the initial velocity is zero and when it reaches the floor its height is zero.

           0 = y₀ +0 - ½ to t²

           a = $\frac{2y_o}{t^2}$

We calculate

           a = $\frac{2 \ 1 }{0.64^2 }$  

           a = 4.88 m / s²

2d part. Let's find the tension in the string.

Let's use Newton's second law for the hanging body.

          W -T = m a

           T = W - ma

           

Body weight is

           W = mg

 

We substitute

            T = m (g-a)

Let's calculate

           T = 1.7 (9.8 -4.88)

           T = 8.364 N

3rd part. We look for the moment of inertia of the pulley.

We use Newton's second law for rotational motion in the pulley.

           τ = I α

           T r = I α

           

Linear and angular quantities are related.

          a = α r

          α = a / r

We substitute

         $T r = I \frac{a}{r} \\I = \frac{T r^2}{a}$

Indicate the mass of the pulley is 2.3 kg and the diameter of the  pulley  is d = 10.4 cm, its radius is half the diameter r = 5.2 cm = 0.052 m.

          I = $\frac{8.364 \ 0.05^2 }{4.88}$  

          I = 4.634 10⁻³ kg m²  

          $I_{real}$  = 4.634 10⁻³ kg m²

4th part. We compare the moments of inertia.

         I = ½ m r²

         I = ½ 2.3 0.052²

         $I_{tabule}$ = 3.11 10-3 kg m²

To compare, let's look for the ratio of the two moments of inertia.

         $\frac{I_{real}}{I_{tabule} } = \frac{4.63}{3.11} \\\frac{I_{real}}{I_{tabule} } = 1.49$  

We can see that the real moment of inertia of the pulley is greater than the tabulated one, for this to happen there must be more mass density towards the periphery since the moment of inertia increases when the mass is at a greater distance from the axis of rotation.  

Conclusion using kinematics and Newton's second law for linear and rotational motions we can find the result for the mass distribution is:

• The mass is distributed more towards the edge of the pulley.

Learn more here: brainly.com/question/14527892