Physics, 20.12.2019 20:31, sbednarek

Two uniform, solid cylinders of radius r and total mass m are connected along their common axis by a short, light rod and rest on a horizontal tabletop (fig. 14.29). a frictionless ring at the center of the rod is attached to a spring with force constant k; the other end of the spring is fixed. the cylinders are pulled to the left a distance x, stretching the spring, and then released from rest. due to friction between the tabletop and the cylinders, the cylinders roll without slipping as they oscillate. show that the motion of the center of mass of the cylinders is simple harmonic, and find its period.

Answers: 2

Show answers

Answers

answered: Aj85537

Explanation:

A) To prove the motion of the center of mass of the cylinders is simple harmonic:

System diagram for given situation is shown in attached Fig. 1

We can prove the motion of the center of mass of the cylinders is simple harmonic if

$a_{x} = -\omega^{2} x$

where aₓ is acceleration when attached cylinders move in horizontal direction:

PROOF:

rotational inertia for cylinders is given as:

$I=\frac{1}{2}MR^{2}$ -----(1)

Newton's second law for angular motion is:

∑τ = Iα ------(2)

For linear motion in horizontal direction it is:

∑Fₓ = Maₓ ------ (3)

By definition of torque:

τ = RF --------(4)

Put (4) and (1) in (2)

$RF=\frac{1}{2}MR^{2}\alpha$

from Fig 3 it can be seen that fs is force by which the cylinders roll without slipping as they oscillate

So above equation becomes

$f_{s}=\frac{1}{2}MR\alpha$ ------ (5)

As angular acceleration is related to linear by:

$a= R\alpha$

Eq (5) becomes

$f_{s}=\frac{1}{2}Ma_{x}$ ---- (6)

aₓ shows displacement in horizontal direction

From (3)

∑Fₓ = Maₓ

Fₓ is sum of fs and restoring force that spring exerts:

$\sum F_{x} = f_{s} - kx$ ----(7)

Put (7) in (3)

$f_{s} - kx = Ma_{x}$ [/tex] -----(8)

Using (6) in (8)

$\frac{1}{2}Ma_{x} - kx =Ma_{x}$

$a_{x} = \frac{2k}{3M} x$ --- (9)

For spring mass system

$a= -\omega^{2} x$ ----- (10)

Equating (9) and (10)

$\omega^{2} = \frac{2k}{3M}$

$\omega = \sqrt{ \frac{2k}{3M}}$

then (9) becomes

$a_{x} = - \omega^{2}x$

(The minus sign says that x and aₓ have opposite directions as shown in fig 3)

This proves that the motion of the center of mass of the cylinders is simple harmonic.

B) Time Period

Time period is related to angular frequency as:

$T=\frac{2\pi }{\omega}$

$T = 2\pi \sqrt{\frac{3M}{2k}$

Two uniform, solid cylinders of radius r and total mass m are connected along their common axis by a

answered: Guest

answer: a ball

explanation: because it will bounce

answered: Guest

An electromagnet is a material in which the current produces electromagnetism and is passed to the core. so to fill in the blanks, choices would be better if there were.an electromagnet is a magnet produced by electric current with a metal core

Other questions on the subject: Physics

Physics, 22.06.2019 00:10, oktacos

The energy released by a chemical reaction can be measured using a calorimeter. when barium hydroxide octahydrate crystals are reacted with dry ammonium chloride inside of a coffee cup calorimeter, the temperature of the 18.00 g of water in the calorimeter decreases from 30.0°c to 8.0°c. the equation for calculating energy absorbed or released by a reaction is: where q is the energy released or absorbed, m is the mass of water in the calorimeter, cp is the specific heat of water, and δt is the observed temperature change. if the specific heat of liquid water is 4.19 j/g·°c, how much energy was absorbed by the reaction?

Answers: 3

continue

Physics, 22.06.2019 10:40, rainbowboi

As you are trying to move a heavy box of mass m, you realize that it is too heavy for you to lift by yourself. there is no one around to , so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. you pull up on the rope to lift the box. use g for the magnitude of the acceleration due to gravity and neglect friction forces. once you have pulled hard enough to start the box moving upward, what is the magnitude f of the upward force you must apply to the rope to start raising the box with constant velocity? express the magnitude of the force in terms of m, the mass of the box.

Answers: 1

continue

Physics, 22.06.2019 12:00, ameliaxbowen7

Suppose a comet has an orbital period of 309.1 years around the sun. what is it’s average distance from the sun?

Answers: 1

continue

Physics, 22.06.2019 12:30, zayeboyd4971

Uppose we consider the system of the three capacitors as a single "equivalent" capacitor. given the charges of the three individual capacitors calculated in the previous part, find the total charge qtot for this equivalent capacitor. express your answer in terms of v and c.

Answers: 2

continue

Do you know the correct answer?

Two uniform, solid cylinders of radius r and total mass m are connected along their common axis by a...