91Ó°ÊÓ

Gravitational force is a fundamental concept in physics that explains why objects are attracted towards each other. This force is responsible for the attraction between the block and the Earth. The gravitational force acting on an object is given by the formula: \[ F_g = Mg \], where \( M \) is the mass of the object and \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth).
In this exercise, we deal with a block of mass \( M \) that is pulled down by the gravitational force. This force is straightforward and doesn't depend on the block's position. Whenever you're dealing with forces, remember that gravity is always working to pull objects towards the Earth.

Work is the energy transferred to or from an object via the application of force along a displacement. The work \( W \) done by a force \( F \) on an object over a displacement \( d \) is given by:\[ W = F \times d \times \text{cos}(\theta) \]However, for forces acting in the direction of the movement (or directly opposite), the cosine factor reduces to 1 or -1. In the given exercise, we compute two types of work:
1. **Work done by the cord's force:**
As the block is lowered at constant acceleration, the cord exerts an upward force (tension) \( T \). The work done by this force is:\[ W_{cord} = -T \times d = -\frac{3Mg}{4} \times d \]2. **Work done by the gravitational force:**
Gravity does work on the descending block and is given by:
\[ W_g = Mg \times d \]
By determining these works independently, we gain insight into how different forces contribute to an object's energy states.

Kinetic energy is the energy an object possesses due to its motion. For an object of mass \( M \) moving at velocity \( v \), the kinetic energy \( KE \) is expressed as:
\[ KE = \frac{1}{2} M v^2 \]
In the exercise, after computing the net work done on the block by considering both the cord's force and gravitational force, we use the work-energy theorem. This theorem states that the net work done on an object equals its change in kinetic energy:
\[ W_{net} = KE \]
So, for the block fallen through distance \( d \), the net work done is:
\[ W_{net} = \frac{Mgd}{4} \]
Thus, the block's kinetic energy is equivalent to this net work. Following this, we solve for the speed \( v \) of the block by rearranging the kinetic energy formula:
\[ \frac{1}{2} M v^2 = \frac{Mgd}{4} \leadsto v = \frac{\text{sqrt}(gd)}{\text{sqrt}(2)} \].

Newton's Second Law is key to understanding the motion of objects under force. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration: \[ F = Ma \]. In the exercise, the block’s constant downward acceleration is given by \( a = \frac{g}{4} \). Using this information, we calculate the tension \( T \) in the cord by applying Newton's second law: \[ T = M(g - a) \], where \( a = \frac{g}{4} \). Thus, \( T = M(g - \frac{g}{4}) = \frac{3Mg}{4} \). Understanding this calculation aids in analyzing how different forces impact the movement, affirming Newton's law as fundamental to mechanics.