What Are the Dynamics of a Collar Sliding on a Rotating Rod?

[negatifzeo]

Homework Statement

A 3-lb collar can slide on a horizontal rod which is free to rotate about a vertical shaft. The collar is intially held at A by a cord attached to the shaft. A spring constant of 2 lb/ft is attached to the collar and to the shaft and is undeformed when the collar is at A. As the rod rotates at the rate ThetaDot=16 rad/s, the cord is cut and the collar moves out along the rod. Neglecting friction and mass of the rod, determine

a)the radial and transverse components of the acceleration at A

b)The acceleration of the collar relative to the rod at A

c)the transverse component of the velocity of the collar at B

Homework Equations

The Attempt at a Solution

I know the solution to the problem. The answers are
a) Ar=0, Atheta=0

b)1536 in./s^2

c)32.0 in/s

I don't feel like this is a difficult problem, but I am definitely missing a key concept. How can you determine these quantities without r as a function of time?

 

[tiny-tim]

Hi negatifzeo!

(have a theta: θ and an omega: ω and try using the X² tag just above the Reply box )

I don't feel like this is a difficult problem, but I am definitely missing a key concept. How can you determine these quantities without r as a function of time?

I suppose you're wondering how you can find r'' without knowing r(t)?

It doesn't matter, because you can work it out from good ol' Newton's second law … F_radial = m(r'' - ω²r)

 

[negatifzeo]

Hi negatifzeo!

(have a theta: θ and an omega: ω and try using the X² tag just above the Reply box )


I suppose you're wondering how you can find r'' without knowing r(t)?

It doesn't matter, because you can work it out from good ol' Newton's second law … F_radial = m(r'' - ω²r)

The angular velocity is given. The mass is given. But we don't know the total force, do we? The total force is broken up into two components, "e-sub-r" and "e-sub-theta", which we do not know.

 

[tiny-tim]

Hi negatifzeo!

(what hapened to that θ i gave you? )

The angular velocity is given. The mass is given. But we don't know the total force, do we? The total force is broken up into two components, "e-sub-r" and "e-sub-theta", which we do not know.

You won't need the e_θ component of the force.

Try it for a) first. ​

 

[DeeNos]

As a scientist, it is important to understand the key concepts and equations involved in solving a problem, rather than just memorizing the solution. In this problem, the key concept is the conservation of energy. The collar is initially at rest at point A, with both potential and kinetic energy equal to zero. When the cord is cut, the collar starts moving along the rod, and energy is conserved as it moves towards point B.

To solve for the radial and transverse components of acceleration at A, we can use the formula for centripetal acceleration, which is given by ac = v^2/r, where v is the velocity and r is the radius. In this case, the collar is initially at rest, so its velocity is zero. The radius, r, is also zero since the collar is at point A. Therefore, the centripetal acceleration at point A is also zero.

To solve for the acceleration of the collar relative to the rod at A, we can use the formula for tangential acceleration, which is given by at = r*alpha, where alpha is the angular acceleration. In this problem, we are given the angular velocity, ThetaDot, which is equal to 16 rad/s. Since there is no friction or mass of the rod, the angular acceleration is also equal to 16 rad/s^2. Thus, the tangential acceleration at point A is 16*0 = 0 in/s^2.

Finally, to solve for the transverse component of the velocity of the collar at point B, we can use the formula for tangential velocity, which is given by v = r*omega, where omega is the angular velocity. In this case, the radius, r, is equal to the length of the rod, which is constant. We know that the angular velocity, ThetaDot, is also constant at 16 rad/s. Therefore, the tangential velocity at point B is equal to 16*length of the rod = 16*12 = 192 in/s.

In summary, by understanding the key concept of conservation of energy and using the appropriate equations, we can solve for the different components of acceleration and velocity in this problem without having to know r as a function of time.