We are given that A is a square matrix of order 3x3 and |A| = -7. We are asked to find the value of the sum: ∑₁³ aᵢ₂ Aᵢ₂, where Aᵢ₂ denotes the cofactor of the element aᵢ₂. This sum corresponds to the determinant of matrix A when expanding along the second column. Specifically, the cofactor expansioRead more
We are given that A is a square matrix of order 3×3 and |A| = -7. We are asked to find the value of the sum:
∑₁³ aᵢ₂ Aᵢ₂, where Aᵢ₂ denotes the cofactor of the element aᵢ₂.
This sum corresponds to the determinant of matrix A when expanding along the second column. Specifically, the cofactor expansion formula for the determinant of a matrix A is:
|A| = ∑ᵢ aᵢⱼ Aᵢⱼ
where aᵢⱼ is the element of the matrix and Aᵢⱼ is its cofactor. Here, the expansion is along the second column. Thus, the sum becomes,
∑₁³ aᵢ₂ Aᵢ₂ = |A|
As we know that |A| = -7, therefore the value of the sum is -7.
To find the work performed in pulling a chain of uniform density onto a table, let us consider a chain 2 meters long, the total mass being 4 kilograms. When put on the table, it overhangs the edge by 60 centimeters. The work performed to raise this overhanging part against the force of gravity can bRead more
To find the work performed in pulling a chain of uniform density onto a table, let us consider a chain 2 meters long, the total mass being 4 kilograms. When put on the table, it overhangs the edge by 60 centimeters. The work performed to raise this overhanging part against the force of gravity can be evaluated using the formula for gravitational potential energy.
Initially, the hanging portion of the chain has a mass that corresponds to its length. Since the entire chain weighs 4 kilograms, the mass of the hanging segment, which is 60 centimeters, is proportionately lighter. As the chain is pulled onto the table, the work required to lift this hanging part involves raising it to a height that gradually reduces to zero.
Since this is the average height of the hanging segment, it is the height to which each mass was lifted to get the work done. The work done against gravity is calculated by multiplying the mass, gravitational force, and height. Using these considerations, the total work done in pulling the whole chain onto the table is found to be about 3.6 joules, which is the energy expended in overcoming the force of gravity in repositioning the chain.
When a machine applies constant power to move a body along a straight line, one may understand the relation of the distance covered by the body with time through the concept of power. Power is the rate of doing work or transferring energy. When the machine gives constant power, it means that the eneRead more
When a machine applies constant power to move a body along a straight line, one may understand the relation of the distance covered by the body with time through the concept of power. Power is the rate of doing work or transferring energy. When the machine gives constant power, it means that the energy being given to the body remains constant with respect to time.
When the body moves, its velocity increases due to the continuous supply of energy. When the velocity of the body increases, the force acting upon it decreases because the product of force and velocity must remain constant in order to keep the power constant. This variation in force and velocity affects the distance traveled by the body over time.
Analysis of the dynamics of the motion, taking place under constant power, reveals that the distance moved by the body is proportional to t³/². The relation can be derived from the basic principles of work, energy, and motion in a system of force, velocity, and time.
Thus, the motion of the body shows how the distance traveled varies as a function of time when constant power is being delivered; it highlights the relationship between energy transfer and motion characteristics.
A couple is the equal and opposite forces applied at different points on an object. These forces produce a rotational effect without causing any translational movement. The forces are in opposite directions but act along parallel lines, and thus result in torque, which produces angular accelerationRead more
A couple is the equal and opposite forces applied at different points on an object. These forces produce a rotational effect without causing any translational movement. The forces are in opposite directions but act along parallel lines, and thus result in torque, which produces angular acceleration in the body. The defining feature of a couple is that forces are equal in magnitude and opposite in direction separated by a fixed distance known as the arm of the couple.
The main effect of a couple on an object is to make it rotate about an axis. Since the forces cancel each other out, the net force acting on the object is zero, so there can be no linear motion. Instead, the couple creates torque, which causes rotation.
To show that the moment of a couple is independent of the axis of rotation selected, note that the torque of a couple does not depend on the choice of axes of rotation within the body. This is because the distance between the lines of action of the two forces is the same, and the forces are equal and opposite. The couple’s moment is thus uniform all over the body, thereby pointing out the inbuilt stability of its rotational effect.
The principle of moments of rotational equilibrium states that for an object to be in a state of rotational equilibrium, the sum of the clockwise moments acting around any axis must equal the sum of the counterclockwise moments around the same axis. In other words, this principle simply states thatRead more
The principle of moments of rotational equilibrium states that for an object to be in a state of rotational equilibrium, the sum of the clockwise moments acting around any axis must equal the sum of the counterclockwise moments around the same axis. In other words, this principle simply states that for an object to remain at rest or to rotate at a constant angular velocity, the torques acting on it must be balanced.
This force applied to the body creates a moment, or torque, in the body due to rotation. The magnitude of each moment will depend on the product of the force applied and the distance of the line of action of that force from the pivot point. If these are not balanced then the object is going to begin rotating.
In rotational equilibrium, all the moments have to cancel out, leading to the total torque acting on the object to be zero. That is why this principle is very important in many applications, including engineering and construction practice, which requires stabilization of structures. For example, a seesaw stays even because both sides of the weights of people placed on it create the same amount of moments. Whenever the side becomes heavy or is shifted further from the pivot, then the seesaw will tip as a sign that equilibrium is lost. Thus, the principle of moments is vital in understanding how balance and stability are achieved in rotational systems.
Given that A = [aᵢⱼ] is a square matrix of order 3 x 3 and |A| = -7, then the value of ∑³₁ ₌ ᵢ aᵢ₂ A ᵢ₂, where Aᵢⱼdemotes the cofactor of element a ᵢⱼis:
We are given that A is a square matrix of order 3x3 and |A| = -7. We are asked to find the value of the sum: ∑₁³ aᵢ₂ Aᵢ₂, where Aᵢ₂ denotes the cofactor of the element aᵢ₂. This sum corresponds to the determinant of matrix A when expanding along the second column. Specifically, the cofactor expansioRead more
We are given that A is a square matrix of order 3×3 and |A| = -7. We are asked to find the value of the sum:
∑₁³ aᵢ₂ Aᵢ₂, where Aᵢ₂ denotes the cofactor of the element aᵢ₂.
This sum corresponds to the determinant of matrix A when expanding along the second column. Specifically, the cofactor expansion formula for the determinant of a matrix A is:
|A| = ∑ᵢ aᵢⱼ Aᵢⱼ
where aᵢⱼ is the element of the matrix and Aᵢⱼ is its cofactor. Here, the expansion is along the second column. Thus, the sum becomes,
∑₁³ aᵢ₂ Aᵢ₂ = |A|
As we know that |A| = -7, therefore the value of the sum is -7.
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A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg. What is the work done in pulling the entire chain on the table?
To find the work performed in pulling a chain of uniform density onto a table, let us consider a chain 2 meters long, the total mass being 4 kilograms. When put on the table, it overhangs the edge by 60 centimeters. The work performed to raise this overhanging part against the force of gravity can bRead more
To find the work performed in pulling a chain of uniform density onto a table, let us consider a chain 2 meters long, the total mass being 4 kilograms. When put on the table, it overhangs the edge by 60 centimeters. The work performed to raise this overhanging part against the force of gravity can be evaluated using the formula for gravitational potential energy.
Initially, the hanging portion of the chain has a mass that corresponds to its length. Since the entire chain weighs 4 kilograms, the mass of the hanging segment, which is 60 centimeters, is proportionately lighter. As the chain is pulled onto the table, the work required to lift this hanging part involves raising it to a height that gradually reduces to zero.
Since this is the average height of the hanging segment, it is the height to which each mass was lifted to get the work done. The work done against gravity is calculated by multiplying the mass, gravitational force, and height. Using these considerations, the total work done in pulling the whole chain onto the table is found to be about 3.6 joules, which is the energy expended in overcoming the force of gravity in repositioning the chain.
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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-5/
A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to
When a machine applies constant power to move a body along a straight line, one may understand the relation of the distance covered by the body with time through the concept of power. Power is the rate of doing work or transferring energy. When the machine gives constant power, it means that the eneRead more
When a machine applies constant power to move a body along a straight line, one may understand the relation of the distance covered by the body with time through the concept of power. Power is the rate of doing work or transferring energy. When the machine gives constant power, it means that the energy being given to the body remains constant with respect to time.
When the body moves, its velocity increases due to the continuous supply of energy. When the velocity of the body increases, the force acting upon it decreases because the product of force and velocity must remain constant in order to keep the power constant. This variation in force and velocity affects the distance traveled by the body over time.
Analysis of the dynamics of the motion, taking place under constant power, reveals that the distance moved by the body is proportional to t³/². The relation can be derived from the basic principles of work, energy, and motion in a system of force, velocity, and time.
Thus, the motion of the body shows how the distance traveled varies as a function of time when constant power is being delivered; it highlights the relationship between energy transfer and motion characteristics.
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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-5/
What is a couple? What effect does it have on a body? Show that the moment of couple is same irrespective of the point of rotation of a body.
A couple is the equal and opposite forces applied at different points on an object. These forces produce a rotational effect without causing any translational movement. The forces are in opposite directions but act along parallel lines, and thus result in torque, which produces angular accelerationRead more
A couple is the equal and opposite forces applied at different points on an object. These forces produce a rotational effect without causing any translational movement. The forces are in opposite directions but act along parallel lines, and thus result in torque, which produces angular acceleration in the body. The defining feature of a couple is that forces are equal in magnitude and opposite in direction separated by a fixed distance known as the arm of the couple.
The main effect of a couple on an object is to make it rotate about an axis. Since the forces cancel each other out, the net force acting on the object is zero, so there can be no linear motion. Instead, the couple creates torque, which causes rotation.
To show that the moment of a couple is independent of the axis of rotation selected, note that the torque of a couple does not depend on the choice of axes of rotation within the body. This is because the distance between the lines of action of the two forces is the same, and the forces are equal and opposite. The couple’s moment is thus uniform all over the body, thereby pointing out the inbuilt stability of its rotational effect.
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See lessState and explain the principle of moments of rotational equilibrium.
The principle of moments of rotational equilibrium states that for an object to be in a state of rotational equilibrium, the sum of the clockwise moments acting around any axis must equal the sum of the counterclockwise moments around the same axis. In other words, this principle simply states thatRead more
The principle of moments of rotational equilibrium states that for an object to be in a state of rotational equilibrium, the sum of the clockwise moments acting around any axis must equal the sum of the counterclockwise moments around the same axis. In other words, this principle simply states that for an object to remain at rest or to rotate at a constant angular velocity, the torques acting on it must be balanced.
This force applied to the body creates a moment, or torque, in the body due to rotation. The magnitude of each moment will depend on the product of the force applied and the distance of the line of action of that force from the pivot point. If these are not balanced then the object is going to begin rotating.
In rotational equilibrium, all the moments have to cancel out, leading to the total torque acting on the object to be zero. That is why this principle is very important in many applications, including engineering and construction practice, which requires stabilization of structures. For example, a seesaw stays even because both sides of the weights of people placed on it create the same amount of moments. Whenever the side becomes heavy or is shifted further from the pivot, then the seesaw will tip as a sign that equilibrium is lost. Thus, the principle of moments is vital in understanding how balance and stability are achieved in rotational systems.
Click here for more: – https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-6/
See less