When a carpenter shuts off his circular saw, the 10.0-inch diameter blade slows from 4670 rpm to zero in 3.00 s.

1) What is the distance traveled by a point on the rim of the blade during the deceleration? (in ft}
2)What is the magnitude of the net displacement of a point on the rim of the blade during the deceleration? (in inch}

1) Well, if we assume that the blade slows down uniformly, we can use the equation d = (v1 + v2) * t / 2, where d is the distance traveled, v1 is the initial velocity, v2 is the final velocity, and t is the time.

Since we want the distance in feet, we need to convert the initial and final velocities from rpm to ft/s. We know that 1 revolution is equal to the circumference of the blade, which is given by C = π * d. So, the initial velocity (v1) is 4670 rpm * C, and the final velocity (v2) is 0 rpm * C (since it slowed down to zero).

Now, plugging in the values, we have d = (4670 * C + 0 * C) * 3.00 s / 2. Since we want the answer in feet, we need to convert the circumference from inches to feet. Remember that there are 12 inches in a foot.

So, the distance traveled by a point on the rim of the blade during the deceleration is approximately (4670 * π * 10.0 * 12) * 3.00 / 2 feet.

2) Now, let's calculate the magnitude of the net displacement. Since the blade goes from 4670 rpm to 0 rpm, we can assume that the direction of displacement is from the initial velocity towards the final velocity.

The magnitude of the net displacement is simply the distance traveled, which we calculated in part 1, so it is approximately (4670 * π * 10.0 * 12) * 3.00 / 2 inches.

To solve these questions, we need to use the formula for the distance traveled during deceleration and the formula for displacement.

1) The distance traveled by a point on the rim of the blade during deceleration can be calculated using the formula:

Distance = (initial velocity + final velocity) * time / 2

In this case, the initial velocity is given as 4670 rpm. To convert this to ft/s, we can use the conversion factor:

1 ft/s = 1.0526 rpm

So, the initial velocity in ft/s is:

4670 rpm * 1.0526 ft/s = 4919.22 ft/s

The final velocity is 0 ft/s because the blade slows down to zero.

The time is given as 3.00 s.

Plugging in these values into the formula:

Distance = (4919.22 ft/s + 0 ft/s) * 3.00 s / 2 = 7378.83 ft

Therefore, the distance traveled by a point on the rim of the blade during deceleration is 7378.83 ft.

2) The magnitude of the net displacement of a point on the rim of the blade during deceleration can be calculated using the formula:

Displacement = (initial velocity + final velocity) / 2 * time

In this case, the initial velocity and final velocity are the same as in question 1.

Plugging in the values:

Displacement = (4919.22 ft/s + 0 ft/s) / 2 * 3.00 s = 7378.83 ft

Therefore, the magnitude of the net displacement of a point on the rim of the blade during deceleration is 7378.83 ft.

To solve these questions, we need to apply the equations of linear motion and rotational motion.

Let's start by finding the angular acceleration (α) of the circular saw blade. The angular acceleration is given by the formula:

α = (ωf - ωi) / t

Where:
α = Angular acceleration
ωf = Final angular velocity (0 rad/s since the blade comes to rest)
ωi = Initial angular velocity (converted to rad/s)
t = Time taken (3.00 s)

The initial angular velocity can be calculated by converting the rpm (revolutions per minute) to rad/s. We know that one revolution is equal to 2π radians, and there are 60 seconds in a minute.

ωi = (4670 rpm) * (2π rad/1 rev) * (1 min/60 s) = 489.8 rad/s

Now we can calculate the angular acceleration:

α = (0 - 489.8 rad/s) / 3.00 s ≈ -163 rad/s^2

1) To find the distance traveled by a point on the rim of the blade during the deceleration, we can use the equation of rotational motion:

θ = ωi * t + 0.5 * α * t^2

Where:
θ = Angle traveled (in radians)
ωi = Initial angular velocity
t = Time taken
α = Angular acceleration

Since the blade comes to rest, the final angular velocity is zero, so we don't include it in the equation.

θ = (489.8 rad/s) * (3.00 s) + 0.5 * (-163 rad/s^2) * (3.00 s)^2

Simplifying the equation, we get:

θ ≈ 1487 rad

To convert this angle to distance in feet, we need to know the circumferential distance traveled by a point on the rim of the blade. The circumference (C) of the circular saw blade can be calculated using the formula:

C = 2π * r

Where:
C = Circumference
r = Radius of the blade (diameter / 2)

Given that the diameter of the blade is 10.0 inches, the radius is 5.0 inches (0.4167 ft). Therefore:

C ≈ 2π * 0.4167 ft ≈ 2.617 ft

To find the distance traveled, we multiply the angle (in radians) by the circumference:

Distance = θ * C ≈ 1487 rad * 2.617 ft ≈ 3890 ft

So, the distance traveled by a point on the rim of the blade during deceleration is approximately 3890 feet.

2) The magnitude of the net displacement is equal to the distance traveled, which we calculated to be 3890 feet. However, since the question asks for the answer in inches, we need to convert the distance to inches.

1 ft = 12 inches, so:

Displacement in inches = 3890 ft * 12 inches/ft ≈ 46680 inches

Therefore, the magnitude of the net displacement of a point on the rim of the blade during deceleration is approximately 46680 inches.

The avg velocity is 4670rpm/2

turns= avgwelocity*time
distance=turns*PI*Diameter
Net displacement?

Even turn=0 displacement. Now, with the remainder, calculate the finalpoint, and the initial point. The difference is displacemetn