The tension in a string from which a 4.0 kg object is suspended in an elevator is equal to 44 N. What is the acceleration of the elevator?

Steps to Solve

1. Identify the forces acting on the object

The object has two main forces acting on it: the gravitational force ($F_g$) and the tension force ($T$).

The gravitational force can be calculated using the formula:

$$ F_g = mg $$

where $m = 4.0 , \text{kg}$ is the mass of the object and $g \approx 9.8 , \text{m/s}^2$ is the acceleration due to gravity.

2. Calculate the gravitational force

Now we can plug in the values to find the gravitational force:

$$ F_g = (4.0 , \text{kg})(9.8 , \text{m/s}^2) = 39.2 , \text{N} $$

3. Apply Newton's second law of motion

According to Newton’s second law, the net force ($F_{net}$) acting on the object can be expressed as:

$$ F_{net} = ma $$

where $a$ is the acceleration. The net force acting on the object is the difference between the tension and the gravitational force when the object is accelerating upwards:

$$ F_{net} = T - F_g $$

4. Set up the equation for acceleration

Now we can set up the equation using Newton's second law. In this case, we can express it as:

$$ T - mg = ma $$

Rearranging gives us:

$$ a = \frac{T - mg}{m} $$

5. Substitute known values and calculate acceleration

If we have a specific value for the tension ($T$), we can substitute that into the equation. For example, let's assume the tension is 50 N:

$$ a = \frac{50 , \text{N} - 39.2 , \text{N}}{4.0 , \text{kg}} = \frac{10.8 , \text{N}}{4.0 , \text{kg}} = 2.7 , \text{m/s}^2 $$