Find the diameter of a pulley which is driven at 360 rpm by a belt moving at 40 ft/s.

Then in 1 s the pulley turns through an angle theta measuring 12 pi radians and a point on the rim travels a distance s= 40 ft.

Circumference = pi*D = 3.14D ft.

Vp = Vb, Vp is velocity of the pulley, and Vb is velocity of the belt.

3.14D ft/rev * 360rev/60s = 40.
D = 2.12 Ft. = Diameter of the pulley.

360 rpm=6 rps

(360rev./min.)(1min./60sec.)=6rps

Well, let's do some calculations while having a little fun! Strap in for the ride, my friend.

Given that the pulley is driven at 360 rpm (rotations per minute), which is the same as 360/60 = 6 rotations per second. Now, we're talking about a pulley here, not a washing machine on a spin cycle!

Since the pulley is driven by a belt moving at 40 ft/s, we can use this information to find the circumference of the pulley. After all, a belt goes all around a pulley, just like a belt holds up my polka dot pants!

The circumference of the pulley can be calculated by dividing the distance traveled by the point on the rim, which in this case is 40 ft, by the angle turned by the pulley, which is 12π radians. So, the circumference of the pulley is 40 ft / (12π radians) = 10/3π ft.

Now, we know that the circumference of a circle is given by the formula: C = πd, where d is the diameter of the circle. Since the circumference of the pulley is 10/3π ft, we can equate it to πd and solve for d.

10/3π = πd

Canceling out the π on both sides, we get:

10/3 = d

So, the diameter of the pulley is 10/3 units. But beware, this is no ordinary-sized pulley - it's more like a circus-sized wheel!

Hope that brings a smile to your face, my friend!

To find the diameter of the pulley, we need to use the formula that relates the linear speed of the belt to the rotational speed of the pulley.

The linear speed of the belt is given as 40 ft/s, and the rotational speed of the pulley is given as 360 rpm (revolutions per minute).

First, we need to convert the rotational speed from rpm to radians per second.

1 revolution = 2π radians
1 minute = 60 seconds

So, 360 rpm = 360 revolutions / 1 minute = 360 * 2π radians / 60 seconds ≈ 37.7 radians / second.

Now, let's use the formula:

Linear speed = Rotational speed * Radius

The linear speed is given as 40 ft/s, and we need to find the radius of the pulley.

Let's rearrange the formula to solve for radius:

Radius = Linear speed / Rotational speed

Substituting the given values:

Radius = 40 ft/s / 37.7 radians/s ≈ 1.061 ft.

Now, we know the radius of the pulley, but we need to find the diameter.

Diameter = 2 * Radius ≈ 2 * 1.061 ft ≈ 2.122 ft.

Therefore, the diameter of the pulley is approximately 2.122 feet.

Regarding the second part of the question:

In 1 second, the pulley turns through an angle theta measuring 12π radians, and a point on the rim travels a distance s = 40 ft.

From the given information, we can see that the distance traveled along the rim of the pulley is equal to the circumference of the pulley.

Circumference = 2π * Radius = 2π * 1.061 ft ≈ 6.665 ft.

Since the circumference of a circle is equal to 2π times the radius, the distance traveled along the rim is also equal to the angle turned multiplied by the radius.

s = theta * Radius

Substituting the given values:

40 ft = 12π radians * 1.061 ft ≈ 40.168 ft.

Therefore, there seems to be an error in the given information or calculations, as the calculated distance traveled (40.168 ft) does not match the given distance traveled (40 ft).

let the diameter be D ft

in one rotation the belt moves Dπ ft.

The wheel rotates at 360 rpm, or at 60 rotations per second
In that 1 second the belt covers 40 ft.
so 60(Dπ) = 40
Dπ = 2/3
D = 2/(3π) ft = appr .2122 ft or appr 2.546 inches