Understanding the Problem: Crane Lifting a Mass

This problem involves a crane lifting a mass with uniform acceleration. We are given the mass of the object, its initial and final velocities, and the time taken. We need to find the tension in the supporting cable during this lifting process.

When an object is lifted by a cable, there are two main vertical forces acting on it: the upward force exerted by the cable (tension, T) and the downward force due to gravity (weight, mg). Since the object is accelerating upwards, there is a net upward force.

Step-by-Step Solution

1. Calculate the Acceleration

The mass starts from rest (\(u = 0\) m/s) and reaches a velocity of \(v = 3\) m/s in \(t = 2\) s. Since the acceleration is uniform, we can use the first equation of kinematics:

\(v = u + at\)

Substituting the given values:

\(3 \text{ m/s} = 0 \text{ m/s} + a \times 2 \text{ s}\)

Solving for acceleration \(a\):

\(a = \frac{3}{2} \text{ m/s}^2 = 1.5 \text{ m/s}^2\)

The acceleration of the mass is 1.5 m/s² upwards.

2. Apply Newton's Second Law

Now we consider the forces acting on the mass. Taking the upward direction as positive, the forces are:

• Tension (T) acting upwards.
• Weight (mg) acting downwards, where \(m = 200\) kg and we take the acceleration due to gravity \(g = 10\) m/s² (a common approximation in such problems).

According to Newton's second law (\(\Sigma F = ma\)), the net force acting on the mass is equal to its mass multiplied by its acceleration:

\(T - mg = ma\)

We want to find the tension T. Rearranging the equation:

\(T = mg + ma\)

3. Substitute Values and Calculate Tension

Substitute the values for mass \(m\), acceleration due to gravity \(g\), and the calculated acceleration \(a\):

\(T = (200 \text{ kg} \times 10 \text{ m/s}^2) + (200 \text{ kg} \times 1.5 \text{ m/s}^2)\)

\(T = 2000 \text{ N} + 300 \text{ N}\)

\(T = 2300 \text{ N}\)

The tension in the supporting cable is 2300 N.

Summary of Calculation

The tension \(T\) is the sum of the weight of the object (\(mg\)) and the force required to accelerate it upwards (\(ma\)).

Conclusion

Based on the calculations using kinematics and Newton's second law, the tension in the supporting cable is 2300 N.

Revision Table: Key Concepts

Additional Information: Lifting with Constant Velocity vs. Acceleration

It's important to understand the difference in tension when lifting at a constant velocity compared to lifting with acceleration.

• Lifting at Constant Velocity: If the crane were lifting the mass at a constant velocity (zero acceleration), Newton's second law would be \(T - mg = m \times 0\), which simplifies to \(T = mg\). In this case, the tension would be equal to the weight (2000 N).
• Lifting with Upward Acceleration: As shown in this problem, when accelerating upwards, the tension \(T\) must be greater than the weight \(mg\) to provide the necessary net upward force \(ma\). The tension is \(T = mg + ma\).
• Lifting with Downward Acceleration: If the mass were accelerating downwards, the net force would be downwards. Taking upward as positive, the equation would be \(T - mg = -ma\), or \(T = mg - ma\). The tension would be less than the weight.

This problem highlights that the force in the cable is not always equal to the object's weight; it depends on the object's acceleration.