Wheel A of radius ra = 13.9 cm is coupled by belt B to wheel C of radius rc = 34.3 cm. Wheel A increases its angular speed from rest at time t = 0 s at a uniform rate of 5.7 rad/s2. At what time will wheel C reach a rotational speed of 168.2 rev/min, assuming the belt does not slip?
The product of rotational speed and radius is the same for each wheel. So is the product of angular acceleration and radius. Use those facts to deduce the time required.
ok..so for c: v = w*r = 168.2*0.343m =57.6926
and therefore for a: w = v/r = 57.6926/0.139m = 415.055
is that correct?
and for the acceleration:
for a: a= 5.7*0.139 = 0.7923
and for c: 0.7923/0.343 = 2.3099
but how do I get time ??
I thought I told you that before.
To get the time, divide the angular velocity change by the angular acceleration
tg chien
To solve this problem, we need to find the time it takes for wheel C to reach a rotational speed of 168.2 rev/min.
First, we need to convert the rotational speed of wheel C to rad/s. Since 1 rev = 2π rad, we can convert 168.2 rev/min to rad/s as follows:
Rotational speed of wheel C = 168.2 rev/min * (2π rad/1 rev) * (1 min/60 s)
= 17.7 rad/s
Now, let's consider the relationship between the angular speed and time for an object undergoing constant angular acceleration. The formula is:
θ = ωi * t + (1/2) * α * t^2
Where:
- θ is the angular displacement
- ωi is the initial angular speed
- α is the angular acceleration
- t is the time
In this problem, wheel A starts from rest (ωi = 0) and has a constant angular acceleration (α = 5.7 rad/s^2). Wheel C reaches a final angular speed of 17.7 rad/s.
We are looking for the time it takes (t) for wheel C to reach this speed. We know the initial angular speed (ωi) and the angular acceleration (α). Rearranging the formula, we get:
θ = (1/2) * α * t^2
Since wheel A and wheel C are coupled by a belt, they have the same angular displacement. Therefore, we can write:
θ = ra * ωa = rc * ωc
Where:
- ra is the radius of wheel A
- rc is the radius of wheel C
- ωa is the final angular speed of wheel A
- ωc is the final angular speed of wheel C
Substituting the values and rearranging the formula, we get:
(1/2) * α * t^2 = (ra * ωa) / rc * ωc
Simplifying further, we get:
t^2 = (2 * ra * ωa) / (rc * ωc * α)
Finally, we can solve for t by taking the square root of both sides:
t = sqrt((2 * ra * ωa) / (rc * ωc * α))
Plugging in the given values, we can calculate the time it takes for wheel C to reach a rotational speed of 168.2 rev/min.