In Fig. 10-28, wheel A of radius rA is coupled by belt B to wheel C of radius rC. The angular speed of wheel A is increased from rest at a constant rate α. Find the time needed for wheel C to reach angular speed ω assuming the belt does not slip. (Hint: If the belt does not slip, the linear speeds at the two rims must be equal.) State your answer in terms of the given variables.
FIGURE: imgur . com/QxZR5. gif
(remove spaces and paste)
I tried:
(rA/rC)*α
and
ωC/αC
but said it was wrong.
To solve this problem, we will use the concept that the linear speed of the rims of wheels A and C must be equal because the belt does not slip.
The linear speed of wheel A is given by vA = rA * α * t, where t is the time.
The linear speed of wheel C is given by vC = rC * ωC, where ωC is the angular speed of wheel C.
Setting these two linear speeds equal:
rA * α * t = rC * ωC
Simplifying and solving for t:
t = (rC * ωC) / (rA * α)
Therefore, the time needed for wheel C to reach angular speed ω is t = (rC * ωC) / (rA * α), where rA, rC, α, and ωC are the given variables.
To find the time needed for wheel C to reach angular speed ω, we need to consider the relationship between the angular speed and radius of each wheel.
Given that the linear speed at the rims of both wheels must be equal (because the belt does not slip), we can use the formula for linear speed:
v = ω * r
Where v is the linear speed, ω is the angular speed, and r is the radius.
For wheel A:
vA = ωA * rA -------(1)
For wheel C:
vC = ωC * rC -------(2)
Since the linear speeds of both rims must be equal, we can equate equations (1) and (2):
ωA * rA = ωC * rC
Now, we are given that wheel A starts from rest and increases its angular speed at a constant rate α. Using the equations of angular motion, we have:
ωA = α * t -------(3)
Where t is the time. Substituting equation (3) into the equation above, we get:
α * rA * t = ωC * rC
Simplifying, we can solve for t:
t = (ωC * rC) / (α * rA)
Therefore, the time needed for wheel C to reach angular speed ω is given by:
t = (ωC * rC) / (α * rA)