Elevator Tension Calculation: Physics Problem

This problem asks us to calculate the tension in the supporting cable of an elevator that is moving upwards with uniform acceleration. We are given the elevator's mass, its final velocity after a certain time, and the acceleration due to gravity.

Information Provided

• Mass of the elevator ($m$): $1000$ kg
• Initial upward velocity ($u$): $0$ m/s (assuming the elevator starts from rest)
• Final upward velocity ($v$): $4$ m/s
• Time interval ($\Delta t$ or $t$): $2$ s
• Acceleration due to gravity ($g$): $9.8$ m/s$^2$

Step 1: Calculate the Elevator's Acceleration

Since the elevator experiences uniform acceleration, we can use a kinematic equation to find the acceleration ($a$). The relevant equation relating final velocity, initial velocity, acceleration, and time is:

$$ v = u + at $$

Plugging in the given values:

$$ 4 \text{ m/s} = 0 \text{ m/s} + a \times (2 \text{ s}) $$

To find the acceleration ($a$), we rearrange the equation:

$$ a = \frac{4 \text{ m/s}}{2 \text{ s}} $$

$$ a = 2 \text{ m/s}^2 $$

So, the elevator is accelerating upwards at a rate of $2$ m/s$^2$.

Step 2: Identify Forces Acting on the Elevator

There are two primary vertical forces acting on the elevator:

• Tension ($T$): This is the upward force exerted by the supporting cable. This is the force we need to find.
• Weight ($W$): This is the downward force due to gravity acting on the elevator's mass.

The weight ($W$) can be calculated using the formula:

$$ W = mg $$

Substituting the given values:

$$ W = (1000 \text{ kg}) \times (9.8 \text{ m/s}^2) $$

$$ W = 9800 \text{ N} $$

Step 3: Apply Newton's Second Law to Find Tension

Newton's Second Law of Motion states that the net force ($F_{net}$) acting on an object is equal to its mass ($m$) multiplied by its acceleration ($a$):

$$ F_{net} = ma $$

In the vertical direction, the net force on the elevator is the difference between the upward tension ($T$) and the downward weight ($W$). Since the elevator is accelerating upwards, the net force is:

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

Equating the two expressions for net force:

$$ T - W = ma $$

Now, we can solve for the tension ($T$):

$$ T = W + ma $$

Substitute the values we know ($W = 9800$ N, $m = 1000$ kg, $a = 2$ m/s$^2$):

$$ T = 9800 \text{ N} + (1000 \text{ kg}) \times (2 \text{ m/s}^2) $$

$$ T = 9800 \text{ N} + 2000 \text{ N} $$

$$ T = 11800 \text{ N} $$

Conclusion

The calculation shows that the tension in the supporting cable required to accelerate the 1000 kg elevator upwards at 2 m/s$^2$ is 11800 N.