Understanding Tension in an Upward Accelerating Lift

This problem involves calculating the tension in a cable supporting a lift that is moving upwards with a constant acceleration. We can use Newton's Second Law of Motion to determine this tension.

Applying Newton's Second Law

Newton's Second Law states that the net force acting on an object is equal to the product of its mass and acceleration ($\Sigma F = ma$).

In the case of the lift moving upwards:

• The forces acting on the lift are the tension (T) in the cable acting upwards and the weight (W) of the lift acting downwards.
• Since the lift is accelerating upwards, the net force is directed upwards.

The equation representing the net force is:

\(\Sigma F = T - W\)

According to Newton's Second Law, this net force equals the mass (m) of the lift multiplied by its upward acceleration (a):

\(T - W = ma\)

Relating Weight, Mass, and Gravity

We know that the weight (W) of an object is the force due to gravity acting on its mass (m). The relationship is given by:

\(W = mg\)

From this, we can express the mass of the lift as:

\(m = \frac{W}{g}\)

Calculating Tension

Now, we substitute the expression for mass (m) back into Newton's Second Law equation for the lift:

\(T - W = \left(\frac{W}{g}\right)a\)

To find the tension (T), we rearrange the equation:

\(T = W + \frac{Wa}{g}\)

Finally, we can factor out W:

\(T = W\left(1 + \frac{a}{g}\right)\)

Conclusion

Therefore, the tension in the cable supporting a lift of weight 'W' with an upward acceleration 'a' is \(W\left(1 + \frac{a}{g}\right)\).