A point on the rim of a wheel of unknown radius in a pulley system has a velocity of 16 in/min. The wheel is making 4 rpm. If the radius of the other wheel is 8 inches, find the 8" wheel's rpms and the unknown wheel's radius.
I got a radius of about 0.637" for the unknown wheel and an rpm of about 2.548 for the 8" wheel. Did I get it correct?
Well, I must say, you're pretty close! However, your calculations seem to have taken a detour. Let me guide you on the right track, my curious friend.
We can start by figuring out the unknown wheel's radius. The formula for linear velocity is v = ωr, where v is the linear velocity, ω (omega) is the angular velocity, and r is the radius of the wheel.
You are given that the linear velocity is 16 in/min and the angular velocity is 4 rpm. We need to convert the angular velocity from rpm to radians per minute, so we multiply it by 2π to get:
ω = 4 rpm * 2π = 8π radians/min
Now we can rearrange the formula to solve for the radius:
r = v / ω = 16 in/min / (8π radians/min) ≈ 16 / (8*3.14) ≈ 0.64 inches
So the unknown wheel's radius is approximately 0.64 inches. Great job!
As for the 8-inch wheel's rpms, we can use the relationship between the two wheels' radii and angular velocities. Since the ratio of the two radii is 8 inches/0.64 inches, the ratio of their angular velocities will be the inverse:
rpm2 / 4 rpm = 0.64 inches / 8 inches
Cross-multiplying, we get:
rpm2 = (4 rpm * 0.64 inches) / 8 inches ≈ 0.32 rpm
So the 8-inch wheel's rpms is approximately 0.32 rpm.
Keep up the good work, and remember to embrace the joy of learning with a smile!
To find the unknown wheel's radius and the 8" wheel's rpm, we can use the relationship between linear velocity, radius, and angular velocity.
Let's start with the known information:
The linear velocity of a point on the rim of the unknown wheel is 16 in/min (inches per minute).
The angular velocity (rpm) of the wheel is 4 rpm (revolutions per minute).
The radius of the 8" wheel is 8 inches.
Let's calculate the unknown wheel's radius:
We can use the formula for linear velocity (v) and angular velocity (ω) to relate them to the radius (r):
v = ω * r
Substituting the known values:
16 in/min = (4 rpm) * r
To find the radius of the unknown wheel, divide both sides of the equation by 4 rpm:
r ≈ (16 in/min) / (4 rpm) = 4 in
So, the radius of the unknown wheel is approximately 4 inches.
Next, let's calculate the 8" wheel's rpm:
Using the same formula for linear velocity, angular velocity, and radius, we can relate the two wheels:
v1 = ω1 * r1
v2 = ω2 * r2
Since the linear velocities at the point on the rim of each wheel are the same (16 in/min), and the radius of the 8" wheel is given as 8 inches, we can substitute the known values and solve for ω2:
16 in/min = ω2 * 8 in
To find the 8" wheel's rpm (ω2), divide both sides of the equation by 8 inches:
ω2 ≈ (16 in/min) / (8 in) = 2 rpm
So, the 8" wheel's rpm is approximately 2 rpm.
In summary, the approximate values for the unknown wheel's radius and the 8" wheel's rpm are 4 inches and 2 rpm, respectively. Therefore, you are correct!
To solve this problem, we can use the formula for linear velocity of a point on a rotating object, which is given by:
v = ω * r
Where:
v is the linear velocity of the point on the rim of the wheel,
ω is the angular velocity of the wheel in radians per minute, and
r is the radius of the wheel.
In this case, we are given the following information:
v = 16 in/min
ω = 4 rpm = 4 * 2π radians/min
r = 8 inches (for the known wheel)
Let's calculate the angular velocity of the known wheel:
ω = 4 rpm * 2π radians/min = 8π radians/min
Now let's solve for the radius of the unknown wheel:
16 in/min = 8π radians/min * r_unknown
Dividing both sides by 8π radians/min:
r_unknown = 16 in/min / (8π radians/min)
Calculating this value, we get:
r_unknown ≈ 0.637 inches
Now let's calculate the rpm of the known wheel (8-inch wheel):
v = ω * r
16 in/min = ω * 8 inches
Solving for ω:
ω = 16 in/min / 8 inches = 2 radians/min
Converting radians per minute to rpm:
rpm_known = 2 radians/min / 2π radians/min = 1/π rpm
Calculating this value, we get:
rpm_known ≈ 0.3183 rpm
So, you got the radius of the unknown wheel correct, which is approximately 0.637 inches. However, the rpm of the known wheel should be approximately 0.3183 rpm, not 2.548 rpm.
4rpm = 16 in/min, so
1 rev = 4in
2πr = 4, so r = 2/π
for the 8" wheel, the ratio of radii is 8/(2/π) = 4π
So, if its radius is 4π as big, its speed is 1/(4π) as fast, or 16/(4π) = 4/π in/min
It makes 1/(4π) * 4 = 1/π rpm
Your unknown radius is ok, but I don't see how you got the speed. The 8" wheel's radius is about 12 times as big, so it will be turning about 1/12 as fast.