an pulls a bucket of water from a well off depth h . If mass of rope is m and bucket is M find the work done by man

To determine the work done by a man pulling a bucket of water from a well of depth \( h \), we need to consider the forces involved and the distance over which these forces act. The work done is essentially the energy transferred when the man pulls the bucket upwards against gravity.

Understanding the Forces at Play

When the man pulls the bucket, he is working against the gravitational force acting on both the bucket and the rope. The total weight that he needs to lift can be calculated as follows:

• Weight of the bucket: This is given by the formula \( W_b = M \cdot g \), where \( M \) is the mass of the bucket and \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
• Weight of the rope: If the rope has a mass \( m \), and it is being pulled up as well, the effective weight of the rope that contributes to the work done will depend on how much of the rope is lifted. When the bucket is at the depth \( h \), the entire length of the rope is being lifted, so the weight of the rope is \( W_r = m \cdot g \).

Calculating Total Weight

The total weight \( W \) that the man is lifting can be expressed as:

W = W_b + W_r = (M + m) \cdot g

Work Done by the Man

The work done \( W_d \) by the man in pulling the bucket up to the surface can be calculated using the formula:

W_d = F \cdot d

Where \( F \) is the total force (weight) and \( d \) is the distance over which the force is applied, which in this case is the depth \( h \) of the well.

Putting It All Together

Substituting the total weight into the work formula gives us:

W_d = (M + m) \cdot g \cdot h

Final Expression

Thus, the work done by the man in pulling the bucket of water from a well of depth \( h \) is:

W_d = (M + m) \cdot g \cdot h

This formula captures the essence of the problem: it accounts for both the mass of the bucket and the mass of the rope, multiplied by the gravitational force and the distance the bucket is lifted. This approach not only provides a clear answer but also illustrates the principles of physics at work in everyday situations.