The system of blocks A and B and two pulleys C and D is
assembled as shown in the figure. Neglecting friction in and
the mass of the pulleys, and assuming that the whole system
is initially at rest, determine the acceleration of block A.
The coefficient of kinetic friction between the surface and
block A is 20%. T2 refers to the cable that holds the 200 kg
mass.
To determine the acceleration of block A, we need to analyze the forces acting on the system.
1. First, let's calculate the tension in the cable T2:
- Block B is connected to pulley C, so the tension in the cable T2 is the same as the weight of block B.
- The weight of block B is given by: Weight = mass × gravity = 200 kg × 9.8 m/s^2.
- Therefore, T2 = 200 kg × 9.8 m/s^2.
2. Next, let's calculate the frictional force acting on block A:
- The coefficient of kinetic friction (μ) between the surface and block A is given as 20% or 0.20.
- The normal force on block A is equal to its weight. Therefore, the normal force is: Normal force = mass of A × gravity = mass of A × 9.8 m/s^2.
- The frictional force is given by: Frictional force = coefficient of kinetic friction × normal force.
- Therefore, the frictional force is: Frictional force = 0.20 × (mass of A × 9.8 m/s^2).
3. Now, let's calculate the net force acting on block A:
- The net force is the difference between the tension force T1 and the frictional force.
- The tension force T1 is connected to pulley D, which is connected to block B. Therefore, T1 is also equal to the weight of block B.
- So, the net force on block A is: Net force = T1 - Frictional force = T2 - Frictional force.
4. Finally, we can calculate the acceleration of block A using Newton's second law (F = ma), where a is the acceleration we want to find:
- The net force acting on block A is equal to block A's mass times its acceleration. Therefore, Net force = (mass of A) × a.
- Equating the net force expression from step 3 to the mass times acceleration expression, we have:
T2 - Frictional force = (mass of A) × a.
- Rearranging the equation to solve for acceleration, we get:
a = (T2 - Frictional force) / (mass of A).
By substituting the values calculated in steps 1 and 2, along with the given mass of block A and the coefficient of kinetic friction, into the final equation, you can determine the acceleration of block A.