The 3.1 , 38-cm-diameter disk in the figure is spinning at 310 . How much friction force must the brake apply to the rim to bring the disk to a halt in 2.6 ?
Express your answer using two significant figures.
To find the friction force required to bring the disk to a halt, we can use the concept of rotational motion. The friction force exerts a torque on the disk, causing it to decelerate.
First, we need to calculate the moment of inertia (I) of the spinning disk. The moment of inertia for a uniform disk rotating about its axis is given by the formula:
I = (1/2) * m * r^2
Where:
m is the mass of the disk
r is the radius of the disk
We are given the diameter of the disk, which is 38 cm. To find the radius, we divide the diameter by 2:
r = 38 cm / 2 = 19 cm = 0.19 m
Since the density of the disk is not given, we cannot directly find its mass. However, we know the disk has a certain rotational speed, given as 310 rev/min. To convert this to radians per second, we use the conversion factor:
1 revolution = 2π radians
Therefore,
310 rev/min * (2π rad/1 rev) * (1 min/60 s) = (310 * 2π) / 60 rad/s
Now that we have the angular velocity (ω) of the disk, we can use the equation:
ω = v / r
Where:
v is the linear velocity of a point on the rim of the disk
We can rearrange this equation to solve for v:
v = ω * r
Plugging in the values:
v = (310 * 2π / 60) * 0.19
Now we have the linear velocity of a point on the disk's rim. The friction force (f) can be found using the formula:
f = μ * m * g
Where:
μ is the coefficient of friction between the disk and the brake
m is the mass of the disk
g is the acceleration due to gravity
We can substitute the value of m into this equation using the formula for the mass of a disk:
m = π * r^2 * h * ρ
Where:
h is the thickness of the disk
ρ is the density of the disk
The thickness of the disk and its density are not given, so unfortunately, we cannot calculate the exact friction force without this information.
However, if we assume that the disk is thin and has a low density, we can estimate the friction force using reasonable values for h and ρ. Once we have these values, we can substitute them into the equation and calculate the friction force.
Keep in mind that the final answer should be expressed using two significant figures.