A 5.00 kg mass is connected by a light cord to a 3.00 kg mass on a smooth surface as shown in the figure. The pulley rotates about a frictionless axle and has a moment of inertia of 0.300 k∙m2 and a radius of 0.500 m. Assuming that the cord does not slip on the pulley, find (a) the acceleration of the two masses
To find the acceleration of the two masses, we can use Newton's second law and the concept of torque. Here's how you can do it step by step:
Step 1: Identify the forces acting on each mass.
- The 5.00 kg mass has its weight acting downward (mg) and tension force (T) from the cord acting upward.
- The 3.00 kg mass has only its weight acting downward (mg).
Step 2: Find the net force acting on each mass.
- For the 5.00 kg mass: The net force is equal to the difference between the tension force and the weight.
Ft - mg = ma1 (where a1 is the acceleration of the 5.00 kg mass)
Rearrange the equation to solve for tension:
Ft = ma1 + mg
- For the 3.00 kg mass: The net force is equal to its weight since there are no other forces acting on it.
Ft = mg (where Ft is the tension force acting on the 3.00 kg mass)
Step 3: Relate the tension force on both masses to find the acceleration.
- The tension force acting on both masses is the same because they are connected by the light cord.
Therefore, we can set the two tension equations equal to each other:
ma1 + mg = mg
Step 4: Solve for acceleration (a1).
- Subtract mg from both sides of the equation:
ma1 = 0
This implies that the acceleration of the 5.00 kg mass, a1, is zero.
- Substitute a1 = 0 into one of the earlier equations to find the tension force (Ft) acting on both masses:
Ft = ma1 + mg
Ft = 0 + mg
Ft = mg
Step 5: Calculate the acceleration of the 3.00 kg mass (a2).
- Now that we know the tension force, we can solve for the acceleration of the 3.00 kg mass using the net force equation:
Ft = ma2
mg = ma2
a2 = g
Therefore, the acceleration of the two masses is "g" (acceleration due to gravity), which in this case is approximately 9.8 m/s², downward for the 3.00 kg mass on the smooth surface. The 5.00 kg mass does not accelerate since its acceleration is zero.