in order to raise a mass of 100kg ,a man of mass 60kg fastens a rope to it and passes the rope over a smooth pulley .he climbs the rope with acceleration 5g/4 relative to the rope .the tension in the rope is?

To determine the tension in the rope when a man of mass 60 kg climbs with an acceleration of 5g/4 relative to the rope while raising a mass of 100 kg, we need to analyze the forces acting on both the man and the mass. Let's break this down step by step.

Understanding the Forces Involved

First, we need to identify the forces acting on both the man and the mass. The key forces include:

• The weight of the man (W_man) = mass × gravity = 60 kg × g
• The weight of the mass being lifted (W_mass) = 100 kg × g
• The tension in the rope (T)

Weight Calculations

Using the acceleration due to gravity (g ≈ 9.81 m/s²), we can calculate:

• W_man = 60 kg × g = 60g
• W_mass = 100 kg × g = 100g

Analyzing the Man's Motion

When the man climbs the rope with an acceleration of 5g/4 relative to the rope, we need to consider his effective acceleration. The total acceleration of the man with respect to the ground is the sum of his climbing acceleration and the acceleration due to the weight of the mass he is lifting.

Let’s denote the acceleration of the man relative to the ground as a_man. Since he climbs with an acceleration of 5g/4 relative to the rope, and the rope is accelerating upwards due to the tension, we can express this as:

a_man = a_rope + (5g/4)

Applying Newton's Second Law

For the man climbing the rope, we can apply Newton's second law:

Net force on the man = T - W_man = m_man × a_man

Substituting the known values:

T - 60g = 60 × (a_rope + 5g/4)

Analyzing the Mass Being Lifted

For the mass being lifted, we also apply Newton's second law:

Net force on the mass = T - W_mass = -m_mass × a_rope

Substituting the known values:

T - 100g = -100 × a_rope

Solving the Equations

Now we have two equations:

1. T - 60g = 60 × (a_rope + 5g/4)
2. T - 100g = -100 × a_rope

From the second equation, we can express T in terms of a_rope:

T = -100 × a_rope + 100g

Now, substitute this expression for T into the first equation:

-100 × a_rope + 100g - 60g = 60 × (a_rope + 5g/4)

Combining like terms gives:

-100a_rope + 40g = 60a_rope + 75g

Rearranging terms leads to:

-160a_rope = 35g

Thus, we find:

a_rope = -35g/160 = -7g/32

Finding the Tension

Now that we have a_rope, we can substitute it back into our expression for T:

T = -100 × (-7g/32) + 100g

T = (7000g/32) + 100g = (7000g/32) + (3200g/32) = (10200g/32)

Finally, simplifying gives:

T = 318.75g

Substituting g ≈ 9.81 m/s², we can find the numerical value of T:

T ≈ 318.75 × 9.81 ≈ 3125.6 N

Therefore, the tension in the rope is approximately 3125.6 N.