You pull downward with a force of 32 N on a rope that passes over a disk-shaped pulley of mass 1.3 kg and radius 0.075 m. The other end of the rope is attached to a 0.60 kg mass. Find the linear acceleration of the 0.60 kg mass.
To find the linear acceleration of the 0.60 kg mass, we need to analyze the forces acting on it.
First, let's consider the force you applied by pulling downward on the rope. This force is 32 N and directed downward.
Next, we need to consider the force due to the tension in the rope. Since the rope is inextensible, the tension in the rope will be the same throughout.
Now, let's consider the pulley. The pulley has both mass and radius, so it will contribute to the overall system's inertia.
To get the linear acceleration, we will use Newton's second law, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration.
Step-by-step solution:
1. Calculate the moment of inertia of the pulley:
- The moment of inertia of a disk about its center is given by the equation I = (1/2) * m * r^2, where "m" is the mass and "r" is the radius.
- In this case, the mass of the pulley is 1.3 kg and its radius is 0.075 m.
- Substituting these values into the equation, we get I = (1/2) * 1.3 kg * (0.075 m)^2.
- Calculate the moment of inertia I.
2. Calculate the tension in the rope:
- The tension in the rope is the force you applied when pulling downward.
- The tension in the rope is also equal to the force of gravity acting on the 0.60 kg mass.
- Calculate the force of gravity acting on the 0.60 kg mass by multiplying its mass by the acceleration due to gravity (9.8 m/s^2).
- Set the tension in the rope equal to the force of gravity and solve for the tension.
3. Set up an equation using Newton's second law:
- The force applied to the system by pulling downward is equal to the tension in the rope.
- This force is also equal to the mass of the 0.60 kg mass multiplied by its acceleration.
- Set the force applied equal to the mass of the 0.60 kg mass multiplied by its acceleration and solve for the acceleration.
4. Now that you have calculated the acceleration, you have found the linear acceleration of the 0.60 kg mass.
In summary, calculate the moment of inertia of the pulley, calculate the tension in the rope, and then use Newton's second law to set up an equation and solve for the acceleration.