A small rubber wheel is used to drive a large pottery wheel, and they are mounted so that their circular edges touch. The small wheel has a radius of 1.7 cm and accelerates at the rate of 6.9 rad/s^2, and it is in contact with the pottery wheel (radius 27.0 cm) without slipping.
(1) Calculate the angular acceleration of the pottery wheel.
Express your answer using two significant figures.
(2) Calculate the time it takes the pottery wheel to reach its required speed of 59 rpm.
The angular acceleration is
(1.7/27)(6.9rad/s^2)=0.434 rad/s^2
The it takes the bigger wheel to reach 59rpm
(59 rev/mn)(2πrad/rev)(mn/60s)= 6.18 rad/s
w = a/t so t =w/a
t= 6.18 rad/s / 0.434rad/s^2
t = 14.2 s
(1) Well, let's get this wheel party started! We can start by finding the angular acceleration of the small rubber wheel. Given that its radius is 1.7 cm and the acceleration is 6.9 rad/s^2, it's like watching a wheel on steroids!
The formula for angular acceleration is just a = α * r, where α is the angular acceleration and r is the radius of the wheel. Plugging in the values, we get:
α = a / r
= 6.9 rad/s^2 / 1.7 cm
≈ 4.06 rad/s^2
So, the angular acceleration of the small wheel is approximately 4.06 rad/s^2.
Now, with their edges touching, these wheels are close pals. Since there's no slipping, the angular acceleration of the small wheel is the same as that of the pottery wheel. So, drumroll please!
The angular acceleration of the pottery wheel is also approximately 4.06 rad/s^2.
(2) Oh, we're still not done! Now let's find out how much time it takes for the pottery wheel to reach its required speed of 59 rpm. We'll have a spin-tastic time!
The required speed is 59 revolutions per minute (rpm). To convert this to radians per second, we can use the conversion factor of 1 revolution = 2π radians. So, the required speed is:
59 rpm * (2π radians / 1 revolution) * (1 minute / 60 seconds)
≈ 6.19 rad/s
Goodness gracious, this pottery wheel is getting quite the spin!
Since we have the angular acceleration (α) and the final angular speed (ω), we can use the formula ω = α * t to find the time (t) it takes to reach that speed. Rearranging the formula gives:
t = ω / α
= 6.19 rad/s / 4.06 rad/s^2
≈ 1.52 s
So, the pottery wheel takes approximately 1.52 seconds to reach its required speed of 59 rpm.
Now that's pottery in a spin!
To solve this problem, we can apply the concept of rotational kinematics and the relationship between the small rubber wheel and the large pottery wheel.
(1) Calculating the angular acceleration of the pottery wheel:
We can use the concept of rotational kinematics, specifically the relationship between angular acceleration (α) and linear acceleration (a) for objects in rolling motion:
α = a / r
Where:
α is the angular acceleration
a is the linear acceleration
r is the radius of the object
In this case, the small rubber wheel has a radius of 1.7 cm, and it has an acceleration of 6.9 rad/s^2. Therefore, the linear acceleration (a) of the rubber wheel is the same as its angular acceleration (α).
The radius of the pottery wheel is given as 27.0 cm. Since the small rubber wheel and the pottery wheel are in contact without slipping, their angular velocities are the same.
α (pottery wheel) = α (rubber wheel) = 6.9 rad/s^2
Therefore, the angular acceleration of the pottery wheel is 6.9 rad/s^2.
(2) Calculating the time it takes for the pottery wheel to reach its required speed of 59 rpm:
First, we need to convert the speed of the pottery wheel from rpm to rad/s.
Speed in rad/s = (speed in rpm) * (2π rad/1 min) * (1 min/60 s)
So, for 59 rpm, the speed of the pottery wheel is:
59 rpm * (2π rad/1 min) * (1 min/60 s) ≈ 6.183 rad/s
Now, we can use the formula for angular acceleration to find the time:
α = Δω / Δt
Where:
α is the angular acceleration
Δω is the change in angular velocity (final angular velocity - initial angular velocity)
Δt is the change in time
Since the angular acceleration (α) of the pottery wheel is constant (from part 1), we can use the following formula:
Δω = α * Δt
We need to find Δt, so we rearrange the equation:
Δt = Δω / α
Plugging in the values:
Δt = 6.183 rad/s / 6.9 rad/s^2 ≈ 0.895 s
Therefore, it takes approximately 0.895 seconds for the pottery wheel to reach its required speed of 59 rpm.
1. 1.7/27 * 6.9 = 0.434 RAD/s^2.
2. V = 59 REV/min * 2pi RAD/REV * 1/60 min/s = 6.18 RAD/s.
V = a * t,
6.18 = 0.434t,
t = 6.18/0.434 = 14.2 s.