PLEASE SOMEONE HELP WITH CORREECT STEPS, I WIL LGIVE 3000 POINTS SHOW ALL YOUR S

Question

PLEASE SOMEONE HELP WITH CORREECT STEPS, I WIL LGIVE 3000 POINTS

SHOW ALL YOUR STEPS PLEASE

1). The pulley below consists of two joined disks turning as one, the larger has a radius 12 cm and the smaller a radius of 6 cm, and moreover, the mass of the pulley is negligible compared to the hanging masses. The hanging masses are m=15kg and M=25kg.What is the acceleration (m/s^2) of the smaller mass, m? The string is thin and has no mass.

answers are:

a)1.51 b)2.45 c.6.03 d.5.88 e.6.53

SHOW ALL YOU STEPS

Explanation / Answer

A pulley of moment of inertia I is made of two disks of radii r1 and r2, welded together. The masses m1 and m2 are attached to two strings, and each string is wound around one of the disks as shown in the figure.

2. Second Newton's law for rotation

The second Newton's law for rotation for the pulley is:
?net = ?? = I ?    (1)
That is :

T1 r1 - T2 r2 = I ?    (1')

? is the angular speed of the pulley. This angular speed ? is the same for the two disks because they are attached to each other.

At the static equilibrium (for rotation and translation), the angular speed is constant or zero (system at rest). Then: ? = d?/dt = 0. Therefore, the formula (1') becomes:
T1 r1 - T2 r2 = I 0 = 0, then T1 r1 = T2 r2

The two torques ?1 = T1 r1 and ?2 = T2 r2 are equal when the system is at rest or at constant speed.

The tangential acceleration "a1" for the disk 1 is
a1 = ? r1    (2)
And
The tangential acceleration "a2" for the disk 2 is
a2 = ? r2    (3)

Multiplying (4') by r1 and (5') by r2, and using the formula (1') yields:
T1 r1 - T2 r2 = r1 m1 (g - a1) - r2 m2(g + a2) = I ?

Using the relationships (2) and (3), we get:
T1 r1 - T2 r2 = r1 m1 (g - ? r1) - r2 m2(g + ? r2) = I ?

Developing:
T1 r1 - T2 r2 = r1 m1 g - ? m1r12 - r2 m2 g - ? m2r22

Rearranging to solve for ? leads to:
r1 m1 g - r2 m2 g - ? (m1r12 + m2r22) = I ?
r1 m1 g - r2 m2 g = ? (m1r12 + m2r22 + I)

Therefore:
? = (r1 m1 g - r2 m2 g)/(m1r12 + m2r22 + I)

We have then:

? = g(r1 m1 - r2 m2)/(m1r12 + m2r22 + I)

a1 = ? r1

a2 = ? r2

T1 = m1(g - a1)

T2 = m2(g + a2)
TO SOLVE THIS PROBLEM MOMENT OF INERTIA IS NEEDED OTHERWISE TAKE I=0

above concept works

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.