Question in rotational motion

[ehabmozart]

Homework Statement

A vacuum cleaner belt is looped over a shaft of radius
0.45 cm and a wheel of radius 1.80 cm. The arrangement of the
belt, shaft, and wheel is similar to that of the chain and sprockets in
Fig. Q9.4. (Just like a bicycle).The motor turns the shaft at 60 rev/s, and the moving
belt turns the wheel, which in turn is connected by another shaft to
the roller that beats the dirt out of the rug being vacuumed. Assume
that the belt doesn’t slip on either the shaft or the wheel. (a) What
is the speed of a point on the belt? (b) What is the angular velocity
of the wheel, in rad/s??

Homework Equations

Rotational motion equation

The Attempt at a Solution

I don't know whether v should be the same for both or w (omega) should be the same for both?

 

[MrWarlock616]

It might be a little easier if you posted some diagrams.

 

[DunWorry]

I think they have the same linear velocity V because if you think about it the belt which is wrapped around these two wheels, the distance traveled on the belt has to be the same for both in the same time interval (otherwise it would fall apart). However the omega is not the same for both, you can see this from V = WR, they have the same V but will have different omega if the radius is different for both wheels. You can also think about it the smaller wheel has to turn more times to get through more of the belt in the same time interval the larger wheel has to turn only a little to get through the same distance of belt.

 

[CWatters]

The speed of all parts of the belt must be the same or it will stretch.

You know the rpm (angular velocity) and dimensions of the small shaft so can write an equation for the speed of a point on it's outer edge. That's the same speed as the belt.

Work same equation backwards to give the angular velocity of the larger pulley.

Edit: If you avoid the temptation to substitute values early on you might stumble on a short cut as a lot will cancel.

 

[Nickie]

I would approach this problem by first identifying the relevant variables and equations for rotational motion. In this case, we have a belt, shaft, wheel, and motor, all of which are involved in the rotation of the system. The relevant equations for rotational motion include the relationship between angular velocity (ω), linear velocity (v), and radius (r):

v = ωr

We also have the equation for angular velocity (ω) in terms of revolutions per second (rev/s):

ω = 2πf = 2πn

Where f is the frequency and n is the number of revolutions per second.

Using this information, we can answer the questions posed in the homework statement:

(a) The speed of a point on the belt will be the same as the linear velocity of the belt, which is given by v = ωr. Therefore, the speed of a point on the belt will be v = (60 rev/s)(0.45 cm) = 27 cm/s.

(b) The angular velocity of the wheel can be found using the equation ω = 2πn, where n is the number of revolutions per second. In this case, the wheel is connected to the belt, which is being turned at 60 rev/s by the motor. Therefore, the angular velocity of the wheel will also be 60 rev/s or 120π rad/s.

It is important to note that the speed and angular velocity are not the same for both the belt and the wheel. The belt is rotating at 60 rev/s, but the wheel is rotating at a different rate due to the difference in radii. This is because the linear velocity of a point on the wheel will be greater than the linear velocity of a point on the belt, since the wheel has a larger radius. However, the angular velocity will be the same for both, as they are connected by the belt and are rotating at the same frequency.