A cable passes over a pulley. Because of friction, the tension in the cable is not eh same on opposite side of the pulley. The force in one side is 110 N, and the force on the other side is 100 N. Assume that the pulley is a uniform disk of mass 2.1kg and radius 0.21. Determine its angular acceleration.
0 0
troque=force*radius =10*angular acceleration
torque=2.1*moment of Inertia*.21
look up moment of inertia for a sold pulley, and solve
To determine the angular acceleration of the pulley, we can use the torque equation: τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The torque acting on the pulley is due to the difference in tension on opposite sides of the pulley, and is given by the equation: τ = (T2 - T1) * r, where T1 is the force on one side of the pulley (100 N), T2 is the force on the other side of the pulley (110 N), and r is the radius of the pulley (0.21 m).
The moment of inertia of a uniform disk can be calculated using the equation: I = (1/2) * m * r^2, where m is the mass of the pulley (2.1 kg), and r is the radius of the pulley (0.21 m).
Substituting the given values into the equations:
τ = (110 N - 100 N) * 0.21 m
= 1 N * 0.21 m
= 0.21 N·m
I = (1/2) * 2.1 kg * (0.21 m)^2
= 0.22 kg·m^2
Now we can calculate the angular acceleration:
0.21 N·m = (0.22 kg·m^2) * α
Simplifying the equation:
α = (0.21 N·m) / (0.22 kg·m^2)
= 0.955 rad/s^2
Therefore, the angular acceleration of the pulley is 0.955 rad/s^2.
To determine the angular acceleration of the pulley, we need to apply Newton's second law for rotational motion, which states that the torque (τ) applied to an object is equal to the moment of inertia (I) multiplied by the angular acceleration (α).
The torque is calculated as the product of the force acting on one side of the pulley (F1) and the radius of the pulley (r). Since the forces on each side of the pulley are not equal due to friction, we can calculate the net torque:
τ = (F1 - F2) * r
First, let's convert the given values into the appropriate units:
F1 = 110 N
F2 = 100 N
r = 0.21 m
Now, substituting these values into the formula:
τ = (110 N - 100 N) * 0.21 m
= 10 N * 0.21 m
= 2.1 Nm
Next, we need the moment of inertia of the pulley. The moment of inertia for a uniform disk rotating around its central axis is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the pulley and r is its radius.
Substituting the given values:
m = 2.1 kg
r = 0.21 m
I = (1/2) * 2.1 kg * (0.21 m)^2
= (1/2) * 2.1 kg * 0.0441 m^2
= 0.04641 kgm^2
Finally, we can calculate the angular acceleration (α) using the formula:
τ = I * α
Rearranging the formula:
α = τ / I
= 2.1 Nm / 0.04641 kgm^2
= 45.27 rad/s^2
Therefore, the angular acceleration of the pulley is approximately 45.27 rad/s^2.