The system shown in the figure below consists of a mass M = 3.5-kg block resting on a frictionless horizontal ledge. This block is attached to a string that passes over a pulley, and the other end of the string is attached to a hanging m = 2.2-kg block. The pulley is a uniform disk of radius 8.0 cm and mass 0.60 kg.
(a) What is the acceleration of each block?
acceleration of M = 3.5 kg m/s2
acceleration of m = 2.2 kg m/s2
(b) What is the tension in the string?
small tension N
large tension N
To find the acceleration of each block in the system, we can use Newton's second law of motion. The equation is given by:
F = ma
Where F is the net force acting on the object, m is the mass of the object, and a is the acceleration of the object. In this case, we can consider the forces acting on each block separately.
For the 3.5 kg block (M):
1. The only force acting on this block is the tension in the string, directed to the right.
2. The net force can be expressed as T - T_friction = ma, where T_friction is the frictional force (which is zero since the surface is frictionless).
3. Therefore, T = ma, where T is the tension in the string and a is the acceleration of the 3.5 kg block.
For the 2.2 kg block (m):
1. The force acting on this block is the tension in the string, directed upward.
2. The weight of the block acts downward, and its value can be calculated as mg, where m is the mass of the block and g is the acceleration due to gravity.
3. The net force can be expressed as T - mg = ma, where T is the tension in the string, m is the mass of the block, g is the acceleration due to gravity, and a is the acceleration of the 2.2 kg block.
Now let's calculate the acceleration of each block:
For the 3.5 kg block (M):
Since the only force acting on it is the tension in the string, we have
T = ma,
where m = 3.5 kg.
Therefore, the acceleration of the 3.5 kg block is equal to the tension divided by its mass:
acceleration of M = T / m = T / 3.5 kg.
For the 2.2 kg block (m):
The net force acting on it is given by
T - mg = ma,
where m = 2.2 kg and g = 9.8 m/s^2.
Rearranging the equation, we have
T = ma + mg,
T = m(a + g).
Therefore, the tension in the string is equal to the sum of the mass multiplied by the sum of the acceleration and acceleration due to gravity:
T = m(a + g) = 2.2 kg (a + 9.8 m/s^2).
The tension in the string is the same for both blocks, so we can equate the two equations for tension and solve for the acceleration:
T / 3.5 kg = 2.2 kg (a + 9.8 m/s^2).
Now, you can solve this equation to find the acceleration of each block.