Work and Energy Concepts

A cord is used to vertically lower a stationary block of mass M at a constant downward acceleration of g/4. The questions are:

1. When the block has fallen a distance d, find: a) The work done by the cord's force on the block. b) The work done by the gravitational force on the block. c) The kinetic energy of the block. d) The speed of the block.

Solution:

This problem is an excellent example to bring about conceptual clarity regarding work, energy, and forces. Let's dive into the solution step-by-step.

Step 1: Understanding the forces acting on the block

We denote the magnitude of the force of the cord on the block as F, which acts upward, opposing the force of gravity and causing an acceleration g/4 downward (instead of g that would have happened if it was not present).

Taking the upward direction as positive, according to Newton's second law: F_net = ma

This gives us: -Mg + F = M(-g/4)

Solving for F: F = (3/4)Mg

So, the force F of the cord is (3/4)Mg, acting upward.

a) Work done by the cord's force

Since the displacement is downward and the force is upwards, the work done by the cord's force should be negative. We can calculate it as:

WF = -F * d

Alternatively, using the formula Work done = F * d * cos(180°) = F * d * (-1):

WF = - (3/4)Mg * d

b) Work done by the gravitational force

The work done by the gravitational force is:

Wg = Mg * d

c) Total work done on the block

The total work done on the block is the sum of the works done by the cord's force and the gravitational force:

Wnet = WF + Wg Wnet = - (3/4)Mg * d + Mg * d Wnet = (1/4)Mg * d

d) Kinetic energy of the block after falling distance d

Using the work-energy theorem, the total work done on the block equals its change in kinetic energy:

Kf - Ki = Wnet

Since the block starts from rest, its initial kinetic energy Ki is zero. Therefore:

Kf = (1/4)Mg * d

e) Speed of the block

Using the kinetic energy formula K = (1/2)Mv² to find the speed v of the block:

(1/2)Mv² = (1/4)Mg * d

Solving for v:

v = √(gd/2)

Key Takeaways:

• The work done by a constant force can be positive or negative depending on the direction of the force relative to the displacement.
• The gravitational force always does positive work when an object moves downward.
• The total work done on an object is the sum of the works done by all forces acting on it.
• The work-energy theorem states that the total work done on an object equals its change in kinetic energy.
• Understanding the forces and the work done by them helps in determining the kinetic energy and speed of the object.