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 tยณ/ยฒ. 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.
Torque can be represented as a vector product of two vectors: the position vector and the force vector. In this context, the torque vector represents the rotational effect of a force applied at a distance from an axis of rotation. The position vector points from the axis of rotation to the point wheRead more
Torque can be represented as a vector product of two vectors: the position vector and the force vector. In this context, the torque vector represents the rotational effect of a force applied at a distance from an axis of rotation. The position vector points from the axis of rotation to the point where the force is applied, while the force vector indicates the direction and magnitude of the applied force.
With torque considered as a vector, both its magnitude and direction are thus reflected. This makes it clearer how a force affects any kind of rotational motion. The magnitude of torque has two factors: the distance from the pivot to the point at which a force is applied as well as the angle through which the force is applied.
The direction of a torque vector is found using the right-hand rule. According to this rule, if you curl the fingers of your right hand in the direction of the force vector while keeping your thumb extended along the position vector, your thumb will point in the direction of the torque vector. This direction indicates the axis of rotation and the sense of rotation that the force will induce on the object. Overall, expressing torque as a vector product simplifies the analysis of rotational dynamics in various physical systems.
The work done by a torque in rotating an object depends on the torque applied and the angular displacement through which the object rotates. If a torque is applied to a body, that body would rotate about an axis. The amount of work done is directly proportional to both the magnitude of the torque anRead more
The work done by a torque in rotating an object depends on the torque applied and the angular displacement through which the object rotates. If a torque is applied to a body, that body would rotate about an axis. The amount of work done is directly proportional to both the magnitude of the torque and the angle through which the object moves. In essence, if a greater torque is applied or if the object rotates through a larger angle, more work is done.
Power, on the other hand, measures the rate at which this work is done. It is defined as the rate at which work is being done over time. For the case of rotational motion, power is related to the work done by the torque and also the time taken to that work. To be more specific, power will be calculated in terms of how much work is done within a certain given time frame when an object is rotated.
This implies that the power developed in a rotating system is dependent on the torque applied to the object as well as its speed of rotation. Understanding torque, work, and power interdependence is thus very important for the analysis of rotational systems and their efficiency in doing work.
Torque in three-dimensional motion refers to the rotational force which is brought about by applying a force at some distance from an axis of rotation. In this sense, torque may be interpreted as the result of a force applied to cause rotation about an axis. To describe torque in three dimensions weRead more
Torque in three-dimensional motion refers to the rotational force which is brought about by applying a force at some distance from an axis of rotation. In this sense, torque may be interpreted as the result of a force applied to cause rotation about an axis. To describe torque in three dimensions we consider the position vector that extends from the axis of rotation to where the force is applied as well as the force vector. The torque, here, can then be represented by its rectangular components along the three axes.
The torque component in the x-direction is given by the product of the y-coordinate of the position vector and the z-component of the force minus the product of the z-coordinate of the position vector and the y-component of the force. The y-component of the torque is the z-coordinate of the position vector multiplied by the x-component of the force, minus the product of the x-coordinate of the position vector and the z-component of the force. Lastly, the z-component of torque comes from the x and y components of the position and force vectors. This approach allows for a detailed analysis of rotational motion in three dimensions.
The magnitude of torque is defined as the product of the magnitude of the force applied and the moment arm, which is the perpendicular distance from the axis of rotation to the line of action of the force. This relationship highlights that the effectiveness of a force in generating rotation dependsRead more
The magnitude of torque is defined as the product of the magnitude of the force applied and the moment arm, which is the perpendicular distance from the axis of rotation to the line of action of the force. This relationship highlights that the effectiveness of a force in generating rotation depends on both the size of the applied force and its distance from the axis. If the force is applied directly at the pivot, then the moment arm is zero, and torque is not produced. However, in the case of application of force with an angle to the pivot, the moment arm can be a maximum, giving a greater torque effect.
Furthermore, only the angular component of the force results in the torque. This is because torque is produced by the force that acts perpendicular to the radius vector, which results in rotation. If a force is applied at an angle to the radius vector, only the component perpendicular contributes to the torque. The component of the force that acts parallel to the radius does not produce rotational motion because it merely pulls or pushes toward the axis without causing rotation. Therefore, understanding the magnitude of the force and the angle at which it is applied is critical in analyzing rotational motion.
1. The gravitational force between two masses is unaffected by the medium between them. 2. Gravitational forces between two bodies are equal and opposite, following Newton's third law of motion. 3. The law of gravitation applies accurately to point masses. 4. Gravitational force acts along the lineRead more
1. The gravitational force between two masses is unaffected by the medium between them.
2. Gravitational forces between two bodies are equal and opposite, following Newton’s third law of motion.
3. The law of gravitation applies accurately to point masses.
4. Gravitational force acts along the line joining two point masses, depending only on the distance (r) without any angular dependence, showing spherical symmetry.
5. Gravitational force is conservative, meaning work done depends only on initial and final positions.
6. The gravitational force between two bodies is not influenced by the presence of other bodies.
In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant ๐บ. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass ๐, attached to the ends of a lightweight rod, forming a dumbbell. This rod iRead more
In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant ๐บ. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass ๐, attached to the ends of a lightweight rod, forming a dumbbell. This rod is suspended vertically by a thin fiber. Two larger lead spheres, each of mass ๐, are placed near the smaller spheres, ensuring all four spheres lie in a horizontal plane. The small spheres are attracted to the larger spheres by the gravitational force, given by:F = G Mm/rยฒ
where ๐ is the distance between the centers of a large sphere and its neighboring small sphere.
Torque and Equilibrium:
The forces on the two small spheres form a couple that exerts a torque on the dumbbell. This torque causes the rod to rotate and twist the suspension fiber until the restoring torque of the fiber balances the gravitational torque. The angle of deflection (๐) is measured using a lamp and scale arrangement, which detects the deflection of a light beam.
Deflecting Torque: tau_deflecting = F . L = G Mm/rยฒ . L,
where ๐ฟ is the length of the rod.
Restoring Torque:
tau_restoring = k๐,
where ๐ is the torsion constant of the fiber (restoring torque per unit angle of twist).
In rotational equilibrium, the two torques are equal and opposite:
G Mm/rยฒ . L = k๐,
Rearranging, the value of ๐บ is:
๐บ = k๐rยฒ / MmL
Determination of ๐บ
By measuring all the quantities on the right-hand side during the experiment, the value of ๐บ can be calculated. Cavendish’s experiment laid the foundation for precise determination of
๐บ, and modern methods have refined its measurement. The currently accepted value is:
G = 6.67 ร 10โปยนยน Nmยฒ kgโปยฒ.
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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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 tยณ/ยฒ. 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.
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See lessExplain how torque can be expressed as a vector product of two vectors. How is the direction of torque determined?
Torque can be represented as a vector product of two vectors: the position vector and the force vector. In this context, the torque vector represents the rotational effect of a force applied at a distance from an axis of rotation. The position vector points from the axis of rotation to the point wheRead more
Torque can be represented as a vector product of two vectors: the position vector and the force vector. In this context, the torque vector represents the rotational effect of a force applied at a distance from an axis of rotation. The position vector points from the axis of rotation to the point where the force is applied, while the force vector indicates the direction and magnitude of the applied force.
With torque considered as a vector, both its magnitude and direction are thus reflected. This makes it clearer how a force affects any kind of rotational motion. The magnitude of torque has two factors: the distance from the pivot to the point at which a force is applied as well as the angle through which the force is applied.
The direction of a torque vector is found using the right-hand rule. According to this rule, if you curl the fingers of your right hand in the direction of the force vector while keeping your thumb extended along the position vector, your thumb will point in the direction of the torque vector. This direction indicates the axis of rotation and the sense of rotation that the force will induce on the object. Overall, expressing torque as a vector product simplifies the analysis of rotational dynamics in various physical systems.
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See lessObtain an expression for the work done by a torque. Hence write the expression for power.
The work done by a torque in rotating an object depends on the torque applied and the angular displacement through which the object rotates. If a torque is applied to a body, that body would rotate about an axis. The amount of work done is directly proportional to both the magnitude of the torque anRead more
The work done by a torque in rotating an object depends on the torque applied and the angular displacement through which the object rotates. If a torque is applied to a body, that body would rotate about an axis. The amount of work done is directly proportional to both the magnitude of the torque and the angle through which the object moves. In essence, if a greater torque is applied or if the object rotates through a larger angle, more work is done.
Power, on the other hand, measures the rate at which this work is done. It is defined as the rate at which work is being done over time. For the case of rotational motion, power is related to the work done by the torque and also the time taken to that work. To be more specific, power will be calculated in terms of how much work is done within a certain given time frame when an object is rotated.
This implies that the power developed in a rotating system is dependent on the torque applied to the object as well as its speed of rotation. Understanding torque, work, and power interdependence is thus very important for the analysis of rotational systems and their efficiency in doing work.
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See lessWrite an expression for torque in three-dimensional motion. Hence write the expressions for the rectangular components of torque.
Torque in three-dimensional motion refers to the rotational force which is brought about by applying a force at some distance from an axis of rotation. In this sense, torque may be interpreted as the result of a force applied to cause rotation about an axis. To describe torque in three dimensions weRead more
Torque in three-dimensional motion refers to the rotational force which is brought about by applying a force at some distance from an axis of rotation. In this sense, torque may be interpreted as the result of a force applied to cause rotation about an axis. To describe torque in three dimensions we consider the position vector that extends from the axis of rotation to where the force is applied as well as the force vector. The torque, here, can then be represented by its rectangular components along the three axes.
The torque component in the x-direction is given by the product of the y-coordinate of the position vector and the z-component of the force minus the product of the z-coordinate of the position vector and the y-component of the force. The y-component of the torque is the z-coordinate of the position vector multiplied by the x-component of the force, minus the product of the x-coordinate of the position vector and the z-component of the force. Lastly, the z-component of torque comes from the x and y components of the position and force vectors. This approach allows for a detailed analysis of rotational motion in three dimensions.
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See lessShow that the magnitude of torque = magnitude of force x moment arm. Also show that only the angular component of the force is responsible for producing torque.
The magnitude of torque is defined as the product of the magnitude of the force applied and the moment arm, which is the perpendicular distance from the axis of rotation to the line of action of the force. This relationship highlights that the effectiveness of a force in generating rotation dependsRead more
The magnitude of torque is defined as the product of the magnitude of the force applied and the moment arm, which is the perpendicular distance from the axis of rotation to the line of action of the force. This relationship highlights that the effectiveness of a force in generating rotation depends on both the size of the applied force and its distance from the axis. If the force is applied directly at the pivot, then the moment arm is zero, and torque is not produced. However, in the case of application of force with an angle to the pivot, the moment arm can be a maximum, giving a greater torque effect.
Furthermore, only the angular component of the force results in the torque. This is because torque is produced by the force that acts perpendicular to the radius vector, which results in rotation. If a force is applied at an angle to the radius vector, only the component perpendicular contributes to the torque. The component of the force that acts parallel to the radius does not produce rotational motion because it merely pulls or pushes toward the axis without causing rotation. Therefore, understanding the magnitude of the force and the angle at which it is applied is critical in analyzing rotational motion.
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See lessMention the characteristic features of gravitational force.
1. The gravitational force between two masses is unaffected by the medium between them. 2. Gravitational forces between two bodies are equal and opposite, following Newton's third law of motion. 3. The law of gravitation applies accurately to point masses. 4. Gravitational force acts along the lineRead more
1. The gravitational force between two masses is unaffected by the medium between them.
2. Gravitational forces between two bodies are equal and opposite, following Newton’s third law of motion.
3. The law of gravitation applies accurately to point masses.
4. Gravitational force acts along the line joining two point masses, depending only on the distance (r) without any angular dependence, showing spherical symmetry.
5. Gravitational force is conservative, meaning work done depends only on initial and final positions.
6. The gravitational force between two bodies is not influenced by the presence of other bodies.
See lessBriefly explain the Cavendish’ s experiment for the determination of the universal constant G.
In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant ๐บ. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass ๐, attached to the ends of a lightweight rod, forming a dumbbell. This rod iRead more
In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant ๐บ. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass ๐, attached to the ends of a lightweight rod, forming a dumbbell. This rod is suspended vertically by a thin fiber. Two larger lead spheres, each of mass ๐, are placed near the smaller spheres, ensuring all four spheres lie in a horizontal plane. The small spheres are attracted to the larger spheres by the gravitational force, given by:F = G Mm/rยฒ
See lesswhere ๐ is the distance between the centers of a large sphere and its neighboring small sphere.
Torque and Equilibrium:
The forces on the two small spheres form a couple that exerts a torque on the dumbbell. This torque causes the rod to rotate and twist the suspension fiber until the restoring torque of the fiber balances the gravitational torque. The angle of deflection (๐) is measured using a lamp and scale arrangement, which detects the deflection of a light beam.
Deflecting Torque: tau_deflecting = F . L = G Mm/rยฒ . L,
where ๐ฟ is the length of the rod.
Restoring Torque:
tau_restoring = k๐,
where ๐ is the torsion constant of the fiber (restoring torque per unit angle of twist).
In rotational equilibrium, the two torques are equal and opposite:
G Mm/rยฒ . L = k๐,
Rearranging, the value of ๐บ is:
๐บ = k๐rยฒ / MmL
Determination of ๐บ
By measuring all the quantities on the right-hand side during the experiment, the value of ๐บ can be calculated. Cavendish’s experiment laid the foundation for precise determination of
๐บ, and modern methods have refined its measurement. The currently accepted value is:
G = 6.67 ร 10โปยนยน Nmยฒ kgโปยฒ.