Question

A cord is used to lower vertically a block of mass \(\mathrm{M}\) by a distance \(\mathrm{d}\) with constant downward acceleration \((9 / 2)\). Work done by the cord on the block is (A) \(-\mathrm{Mgd} / 2\) (B) \(\mathrm{Mgd} / 4\) (C) \(-3 \mathrm{Mgd} / 4\) (D) \(\mathrm{Mgd}\)

Short answer

The work done by the cord on the block is \(-\frac{Mgd}{2}\).

Step-by-step solution

Identify the forces acting on the block

Let's identify the forces acting on the block: 1. Gravitational force (downward) is equal to Mg. 2. Tension force in the cord acts upward. Since the block is having constant downward acceleration of (9/2), the net force (F_net) acting on the block is also downward.

Find the net force on the block

Using Newton's second law for the motion of the block, we have \(F_{net} = Ma\), where M is the mass of the block, and a is the downward acceleration. So, \(F_{net} = M \times \frac{9}{2}\)

Resolve the forces for the block

To find the tension force (T) in the cord, we'll use the following equation: \(F_{net} = T - F_g\) Where \(F_g\) is the gravitational force (Mg) acting on the block. Replacing the net force and the gravitational force in the equation, we get: \(M \times \frac{9}{2} = T - Mg\) Solving for T, we get: \(T = M \times (\frac{9}{2} + g)\)

Calculate the work done by the cord

By definition, work (W) is given by the equation \(W = Fd\cos\theta\), where F is the force applied, d is the displacement, and \(\theta\) is the angle between the force and displacement. Since the tension in the cord opposes the displacement of the block (acts upwards), the angle between the tension force and the displacement is 180 degrees. Therefore, the value of \(\cos\theta = -1\). To find the work done by the cord (W_cord), we'll use tension force (T) found in step 3: \(W_{cord} = T \times d \times (-1)\) \(W_{cord} = -M \times (\frac{9}{2} + g) \times d\) Substituting the value of g: \(W_{cord} = -M \times (\frac{9}{2} + 9) \times d\) Simplifying the equation, we get: \(W_{cord} = -\frac{Mgd}{2}\) The work done by the cord on the block is option (A) \(-\frac{Mgd}{2}\).

Key concepts

Newton's second law

Newton's second law is a fundamental principle in physics that relates the force acting on an object to its acceleration and mass. According to this law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration, expressed as:\[ F_{net} = Ma \].
This equation helps us understand how different forces result in acceleration. For example, if more force is applied to an object, it will accelerate more, assuming its mass stays the same. In the exercise, we see that the block accelerates downward due to the forces acting on it. Remember that the direction of the net force determines if the acceleration is positive or negative. This law is crucial for predicting the motion of objects under various forces.

gravitational force

Gravitational force is a natural phenomenon by which all things with mass are brought toward one another. For objects near the Earth's surface, this force can be calculated as the product of the object's mass and the acceleration due to gravity (g), which is approximately 9.8 m/s². Thus, the gravitational force (\( F_g \)) acting on an object is given by:\[ F_g = Mg \],
where \( M \) is the mass of the object.
In the problem, gravitational force pulls the block downward, contributing to its motion. Understanding gravitational force is important because it affects every object on Earth, keeping us grounded and influencing how objects move.

tension force

Tension force is the force transmitted through a string, rope, or cord when it is pulled tight by forces acting from opposite ends. In physics problems like the one given, tension can support objects against gravity or provide an upward force. The exercise involves calculating tension in the cord that lowers a block with mass. To find tension, use the equation:\[ F_{net} = T - F_g \],
where \( T \) is the tension force. By understanding how tension equilibrates with other forces, we can ensure that objects move as intended without snapping the cord or having unexpected accelerations. It is also a key factor in calculating work done by a force when the object moves.

constant acceleration

Constant acceleration occurs when an object's velocity changes by the same amount in each successive unit of time. This simplifies calculations because the acceleration value remains the same, making predictions about motion more straightforward. In the current exercise, the block descends with a constant acceleration of \( 9/2 \), implying that the forces involved create a steady change in velocity. Constant acceleration problems often use the kinematic equations to link acceleration, velocity, and displacement. This scenario demonstrates how combining constant acceleration with Newton's second law allows prediction of the work done, allowing for practical applications in engineering and mechanics.