The 1.4 kg, 40-cm-diameter disk in the figure below is spinning at 260 rpm. How much friction force must the brake apply to the rim to bring the disk to a halt in 4.5 s?
torque=momentinertia*ang acceleration
look up the momentInertia for a disk
ang acceleration=(260*2PI)/4.5
torque= frictionforce*radius
To find the friction force required to bring the disk to a halt, we can use the concepts of torque and rotational inertia.
First, let's find the rotational inertia of the disk. The rotational inertia for a uniform disk rotating about its central axis can be calculated using the formula:
I = (1/2) * m * r^2
where I is the rotational inertia, m is the mass of the disk, and r is the radius of the disk.
Given:
Mass of the disk (m) = 1.4 kg
Diameter of the disk (d) = 40 cm = 0.4 m
Radius of the disk (r) = 0.4 m / 2 = 0.2 m
Plugging the values into the formula, we can calculate the rotational inertia:
I = (1/2) * 1.4 kg * (0.2 m)^2
I = 0.04 kg*m^2
Now, let's find the angular deceleration of the disk. We know that the angular velocity (ω) is initially 260 rpm. To convert the angular velocity to radians per second, we use the conversion factor:
1 rpm = (2π/60) rad/s
So, the initial angular velocity is given by:
ω = 260 rpm * (2π/60) rad/s
ω = 27.33 rad/s
To calculate the angular deceleration (α), we can use the following formula:
α = (ω - 0) / t
where α is the angular deceleration, ω is the initial angular velocity, and t is the time taken to come to a halt.
Given:
Initial angular velocity (ω) = 27.33 rad/s
Time taken to come to a halt (t) = 4.5 s
Plugging the values into the formula, we can calculate the angular deceleration:
α = (27.33 rad/s - 0) / 4.5 s
α = 6.07 rad/s^2
Now, let's find the torque required to produce this angular deceleration. The torque (τ) can be calculated using the formula:
τ = I * α
where τ is the torque and I is the rotational inertia.
Plugging the values into the formula, we can calculate the torque:
τ = 0.04 kg*m^2 * 6.07 rad/s^2
τ = 0.2428 N*m
Finally, for a disk, the torque exerted by friction is equal to the product of the applied force (F) and the radius (r) of the disk. So, we can find the friction force (F) using the formula:
F = τ / r
Given:
Radius of the disk (r) = 0.2 m
Torque (τ) = 0.2428 N*m
Plugging the values into the formula, we can calculate the friction force:
F = 0.2428 N*m / 0.2 m
F = 1.214 N
Therefore, the brake must apply a friction force of approximately 1.214 N to bring the disk to a halt in 4.5 s.