Two objects (41.0 and 20.8 kg) are connected by a massless string that passes over a massless, frictionless pulley. The pulley hangs from the ceiling. Find the acceleration of the objects and the tension in the string.
To find the acceleration of the objects and the tension in the string, we need to use Newton's second law of motion and consider the forces acting on each object.
Let's start with the equations of motion for each object:
For the 41.0 kg object:
m₁ * a = T - m₁ * g -- Equation 1
For the 20.8 kg object:
m₂ * a = m₂ * g - T -- Equation 2
where:
m₁ = mass of the 41.0 kg object
m₂ = mass of the 20.8 kg object
a = acceleration of the objects
T = tension in the string
g = acceleration due to gravity (9.8 m/s²)
Now we have two equations with two unknowns, a and T. We can solve them simultaneously to find the values.
First, let's solve for tension (T):
From Equation 1:
T = m₁ * (a + g)
Now, substitute this equation for T in Equation 2:
m₂ * a = m₂ * g - m₁ * (a + g)
Next, simplify the equation:
m₂ * a = m₂ * g - m₁ * a - m₁ * g
m₂ * a + m₁ * a = m₂ * g - m₁ * g
(a * (m₁ + m₂)) = (m₂ - m₁) * g
Finally, solve for acceleration (a):
a = ((m₂ - m₁) * g) / (m₁ + m₂)
Using the given values of m₁ = 41.0 kg and m₂ = 20.8 kg, and the acceleration due to gravity g = 9.8 m/s², substitute these values into the equation to calculate the acceleration:
a = ((20.8 kg - 41.0 kg) * 9.8 m/s²) / (41.0 kg + 20.8 kg)
a = (-20.2 kg * 9.8 m/s²) / (61.8 kg)
a ≈ -3.21 m/s²
The negative sign indicates that the objects are moving in the opposite directions.
Now, substitute the calculated acceleration into Equation 1 to find the tension (T):
T = m₁ * (a + g)
T = 41.0 kg * (-3.21 m/s² + 9.8 m/s²)
T ≈ 248.7 N
So, the acceleration of the objects is approximately -3.21 m/s² (opposite directions), and the tension in the string is approximately 248.7 N.