A disk with a mass of 18 kg, a diameter of 50 cm, and a thickness of 8 cm is mounted on a rough horizontal axle as shown on the left in the figure. (There is a friction force between the axle and the disk.) The disk is initially at rest. A constant force, F = 70 N,is applied to the edge of the disk at an angle of 37°, as shown on the right in the figure. After 2.0 s, the force is reduced to F = 27 N,and the disk spins with a constant angular velocity.
How do I find the angular velocity and kinetic energy after 2 seconds?
To find the angular velocity and kinetic energy of the disk after 2 seconds, we can use the principles of rotational motion and energy.
Step 1: Calculate the torque applied to the disk:
The torque is given by the formula: Torque = Force x Lever Arm
The lever arm is the perpendicular distance from the axis of rotation to the point where the force is applied. In this case, the force is applied at the edge of the disk.
Since the force is applied at an angle of 37°, we need to find the component of the force that acts perpendicular to the lever arm.
The perpendicular component of the force is F_perpendicular = F x sin(angle).
Therefore, the torque is Torque = F_perpendicular x r, where r is the radius of the disk.
Step 2: Find the moment of inertia of the disk:
The moment of inertia, I, depends on the shape and mass distribution of the object. For a disk rotating about its central axis, the moment of inertia is given by the formula: I = (1/2) x m x r^2, where m is the mass of the disk and r is the radius of the disk.
Step 3: Use the torque and moment of inertia to find the angular acceleration:
The angular acceleration, α, is related to the torque and moment of inertia by the equation: Torque = I x α.
Therefore, α = Torque / I.
Step 4: Calculate the angular velocity after 2 seconds:
The equation for rotational motion is: ω = ω0 + α x t, where ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.
Since the disk is initially at rest (ω0 = 0), the equation simplifies to: ω = α x t.
Step 5: Calculate the kinetic energy of the disk:
The kinetic energy, KE, is related to the moment of inertia and angular velocity by the equation: KE = (1/2) x I x ω^2.
Let's calculate the angular velocity and kinetic energy after 2 seconds using the given data.
To find the angular velocity and kinetic energy of the disk after 2 seconds, we can use the principles of rotational motion.
First, let's find the moment of inertia of the disk. The moment of inertia, symbolized as I, is a property that quantifies how mass is distributed in an object. For a solid disk rotating around its central axis, the moment of inertia can be calculated using the formula:
I = (1/2) * m * r^2
where m is the mass of the disk and r is its radius. In this case, the mass of the disk is given as 18 kg, and the radius (half the diameter) is 0.5 m. Plugging these values into the formula, we get:
I = (1/2) * 18 kg * (0.5 m)^2
Next, let's find the angular acceleration of the disk. The angular acceleration, symbolized as α, represents how quickly the angular velocity changes. It can be calculated using the formula:
α = τ / I
where τ is the torque applied to the disk and I is the moment of inertia. The torque is given by the product of the applied force and the lever arm, which is the perpendicular distance between the force and the axis of rotation. In this case, the applied force is 70 N, and the angle between the force and the lever arm is 37°. The lever arm can be calculated as:
r' = r * sin(37°)
Plugging the values into the formula, we get:
τ = F * r'
= 70 N * (0.5 m * sin(37°))
Now, we can calculate the angular acceleration:
α = τ / I
Now, set up a system of equations:
70 N * (0.5 m * sin(37°)) = (1/2) * 18 kg * (0.5 m)^2 * α
Evaluate the left-hand side of the equation to find τ:
70 N * (0.5 m * sin(37°)) = 17.5 N * 0.36397
= 6.37595 N·m
Plugging the moment of inertia and torque into the formula for angular acceleration, solve for α:
6.37595 N·m = (1/2) * 18 kg * (0.5 m)^2 * α
Finally, we have the angular acceleration, α. Now, to find the angular velocity (ω) after 2.0 seconds, we can use the equation:
ω = ω0 + α * t
where ω0 is the initial angular velocity (which is 0 in this case), α is the angular acceleration, and t is the time interval. Plugging in the values, we have:
ω = 0 + 6.37595 N·m / ((1/2) * 18 kg * (0.5 m)^2) * 2.0 s
= 1.0 rad/s
The angular velocity after 2.0 seconds is 1.0 rad/s.
To find the kinetic energy (KE) of the disk after 2 seconds, we can use the formula:
KE = (1/2) * I * ω^2
where I is the moment of inertia and ω is the angular velocity. Plugging in the values, we have:
KE = (1/2) * 18 kg * (0.5 m)^2 * (1.0 rad/s)^2
= 0.45 J
The kinetic energy of the disk after 2 seconds is 0.45 Joules.