Definition of Strain: Strain is a measure of deformation, which gives the displacement between the particles in a material body. Strain is defined as the ratio of change in dimension to the original dimension of the material. In mathematical terms, strain (ε) is expressed as: ε = ΔL / L₀ Here, ΔLRead more
Definition of Strain:
Strain is a measure of deformation, which gives the displacement between the particles in a material body. Strain is defined as the ratio of change in dimension to the original dimension of the material. In mathematical terms, strain (ε) is expressed as:
ε = ΔL / L₀
Here, ΔL represents the change in length and L₀ represents the original length.
Why Strain Has No Units and Dimensions:
Strain is a dimensionless quantity because it is a ratio of two lengths (change in length to original length), which means that both numerator and denominator have the same units. Hence, the units cancel out, and one gets a unitless value. Since it is dimensionless, therefore it has no dimensions in the context of physical measurement.
Different Types of Strain:
1. Tensile Strain: It takes place when a material is stretched. It is defined as the change in length divided by the original length (elongation).
2. Compressive Strain: It takes place when a material is compressed. It is defined as the change in length divided by the original length (shortening).
3. Shear Strain: It is that strain which arises due to shear forces applied on a material. It is defined as the change in angle between two lines originally at right angles, expressed as the ratio of lateral displacement to the original length.
4. Volumetric Strain: It is the change in volume per unit original volume of a material, occurring under uniform pressure.
These types of strain are important for understanding how materials respond to various forces and deformations.
Definition of Stress: Stress is defined as the force exerted per unit area within materials. It results from external forces applied to an object, resulting in deformation. Mathematically, stress is represented as σ = F / A where σ is the stress F is the applied force and A is the cross-sectional arRead more
Definition of Stress:
Stress is defined as the force exerted per unit area within materials. It results from external forces applied to an object, resulting in deformation. Mathematically, stress is represented as
σ = F / A
where σ is the stress F is the applied force and A is the cross-sectional area over which the force is applied.
Units:
The SI unit of stress is Pascal (Pa), which is equivalent to Newton per square meter (N/m²).
Dimensions:
The dimensions of stress are represented as [M L⁻² T⁻²], where M is mass, L is length, and T is time.
Different Types of Stress:
1. Tensile Stress: This takes place when any material undergoes stretching or pull forces. The force per unit area along the direction of an applied force defines it.
2. Compressive Stress: It arises when compressive forces compress or squeeze the material. Force per unit area in a direction opposite to an applied force is defined by compressive stress.
3. Shear Stress: It happens when a material is subjected to forces that cause one layer of the material to slide over another. It is the force applied parallel to the surface divided by the area of the surface.
4. Volumetric Stress: This stress is due to changes in volume because of uniform pressure applied in all directions. It is significant in fluid mechanics and materials science.
Understanding these different types of stress is important in analyzing material behavior under various loading conditions.
Elasticity of Solids: Elastic behavior refers to the phenomenon of an ability to regain the original configuration shape and size when the applied external force is removed. This is critical in understanding how a material deforms and recovers under stress. The mechanical spring-ball model is a usefRead more
Elasticity of Solids:
Elastic behavior refers to the phenomenon of an ability to regain the original configuration shape and size when the applied external force is removed. This is critical in understanding how a material deforms and recovers under stress. The mechanical spring-ball model is a useful representation to explain this concept.
Mechanical Spring-Ball Model:
1. Basic Concept: In a mechanical spring-ball model, atoms or molecules in a solid can be visualized as balls held together by springs. The springs are equivalent to the interatomic forces or bonding forces between the atoms.
2. Elastic Deformation: When external force is applied to the solid, balls go a little away from their equilibrium positions; as a result springs stretch or compress because of which elastic deformation takes place. In this type of deformation, the shape of the material changes, but the material remains intact.
3. Restoring Forces: When the applied force is withdrawn, the springs exert restoring forces that restore the balls to their original positions. This is because the interatomic forces are elastic in nature; the material can return to its original shape and size.
4. Elastic Limit: Elastic behavior is seen up to a certain limit called the elastic limit. If the applied force exceeds this limit, then the springs might get permanently deformed and the deformation will be plastic. In this case, the solid cannot regain its original shape.
5. Mathematical Representation: This elastic region has a relationship that can be stated using Hooke’s Law that relates stress as proportional to strain (σ = Eε, where E is the modulus of elasticity).
In this mechanical model of a spring-ball, interatomic forces and arrangements of atoms come into play about deformation and recovery characteristics of the solids under stress.
Deforming Force: A deforming force is any force applied on the outside to change the shape or size of a material. Such forces can result in either elastic or plastic deformation depending on the intensity of the applied force and material characteristics. Deforming forces can occur from tension, comRead more
Deforming Force: A deforming force is any force applied on the outside to change the shape or size of a material. Such forces can result in either elastic or plastic deformation depending on the intensity of the applied force and material characteristics. Deforming forces can occur from tension, compression, shear, or torsion.
Elasticity:
Elasticity refers to a material’s ability to regain its original shape and size after removal of the applied deforming force. Some materials exhibit this so that they can be deformable in a temporary manner while characteristically, stress and strain are directly proportional within the elastic limit. Restoration to the original state is made possible by interatomic forces.
Plasticity:
Plasticity is the ability of a material to undergo permanent deformation once it has been subjected to a deforming force greater than its elastic limit. Once the force is removed, the material will not return to its original shape and will have altered its structure. Plastic behavior is characterized by a lack of proportionality between stress and strain once the elastic limit has been reached.
Perfectly Elastic Bodies:
Perfectly elastic bodies are materials that can return to their original shape and size after the removal of any applied deforming force, regardless of the magnitude of the force, as long as it does not exceed the material’s elastic limit. They exhibit linear stress-strain behavior and follow Hooke’s Law throughout their entire range of deformation.
Example: An ideal rubber band behaves like a perfectly elastic body, which means it stretches and returns to its original shape once the deforming force is removed.
Perfectly Plastic Bodies:
Perfectly plastic bodies are the ones that fail to regain the shape once they lose the externally applied deforming force. Thus, they go for permanent deformation with no evidence of elastic deformation. After yield point, more stress will create only plastic deformation with no further rise in the level of stress.
Example: Modeling clay is an example of a perfectly plastic body, as it can be easily shaped and will retain the new shape after the applied force is removed.
Understanding these concepts is essential for studying material behavior under various forces and applications in engineering and materials science.
Definition of Compressibility: Compressibility is a measure of the ability of a substance to decrease in volume under pressure. It quantifies how much a material will compress when subjected to an applied external force. The compressibility (β) of a substance is defined as the fractional change in vRead more
Definition of Compressibility:
Compressibility is a measure of the ability of a substance to decrease in volume under pressure. It quantifies how much a material will compress when subjected to an applied external force. The compressibility (β) of a substance is defined as the fractional change in volume per unit increase in pressure.
Mathematical Expression:
Compressibility is mathematically expressed as:
β = – (1/V) * (ΔV/ΔP)
where:
– β is the compressibility,
– V is the initial volume,
– ΔV is the change in volume,
– ΔP is the change in pressure.
Units:
The SI unit of compressibility is the reciprocal of pressure which is usually expressed in terms of:
– Pa⁻¹ (Pascal inverse) or
– N⁻¹ m² (Newton inverse per square meter).
Dimensions:
The dimensions for compressibility can be written as:
[M⁻¹ L³ T²]
Where:
M is mass,
L is length,
T is time.
In various disciplines such as fluid mechanics, material science, and engineering, it is highly relevant to know the compressibility because gases and liquids often react to varying conditions of pressure.
Definition of Bulk Modulus of Elasticity: The bulk modulus of elasticity is the coefficient of a medium's resistance toward uniform compression, defined as a ratio of relative change in volume to the intensity of pressure by which the material volume is decreased or increased. For mathematical expreRead more
Definition of Bulk Modulus of Elasticity:
The bulk modulus of elasticity is the coefficient of a medium’s resistance toward uniform compression, defined as a ratio of relative change in volume to the intensity of pressure by which the material volume is decreased or increased. For mathematical expression the following is employed:
K=-ΔP/(ΔV\V)
where K is the bulk modulus ΔP is the change in pressure ΔV is the change in volume and V is the original volume.
Units:
The SI unit of bulk modulus is Pascal (Pa), which is equal to Newton per square meter (N/m²).
Dimensions:
The dimensions of bulk modulus are expressed as [M L⁻¹ T⁻²], where M is mass L is length and T is time.
Definition of Modulus of Elasticity: Modulus of elasticity is a measure of the ability of a material to deform elastically under the influence of a force. It is a measure of the ratio of stress (force per unit area) to strain (deformation) in a material. The modulus indicates how much a material wilRead more
Definition of Modulus of Elasticity:
Modulus of elasticity is a measure of the ability of a material to deform elastically under the influence of a force. It is a measure of the ratio of stress (force per unit area) to strain (deformation) in a material. The modulus indicates how much a material will deform under a given load.
Units:
The SI unit of modulus of elasticity is Pascal (Pa), which is equal to Newton per square meter (N/m²).
Dimensions:
The modulus of elasticity is expressed in units of [M L⁻¹ T⁻²], where M stands for mass, L for length, and T for time.
Some Common Moduli of Elasticity:
1. Young’s Modulus (E): It is the tensile or compressive elasticity of a material, that is, the ratio of tensile stress to tensile strain.
2. Bulk Modulus (K): It represents the resistance offered by a material to uniform compression. It is defined as the ratio of the change in pressure to the relative decrease in volume.
3. Shear Modulus (G): Also known as the modulus of rigidity, it measures a material’s response to shear stress. It is defined as the ratio of shear stress to shear strain.
4. Poisson’s Ratio (ν): It is a measure of the ratio of transverse strain to axial strain in a material subjected to axial stress, but not a modulus in the strict sense.
Hooke’s Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, provided the elastic limit is not exceeded. Mathematically, it is expressed as: F = -kx where F is the restoring force exerted by the spring k is the spring constant andRead more
Hooke’s Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, provided the elastic limit is not exceeded. Mathematically, it is expressed as:
F = -kx
where F is the restoring force exerted by the spring k is the spring constant and x is the displacement from the equilibrium position.
Experimental Verification:
To verify Hooke’s Law experimentally, the following steps can be followed:
1. Setup: Attach a spring vertically to a fixed support and hang weights from the free end of the spring. Use a ruler or measuring tape to measure the displacement.
2. Apply Weights: Gradually add known weights to the spring and measure the extension (displacement) of the spring each time a weight is added. Ensure that the weights are added incrementally and the spring is not overstretched.
3. Record Data: Record the weight applied (force) and the corresponding displacement of the spring.
4. Plotting Graph: Plot a graph of the applied force (y-axis) against the displacement (x-axis). According to Hooke’s Law, the graph should be a straight line passing through the origin, indicating that the force is directly proportional to the displacement.
5. Determine the Spring Constant: The slope of the linear graph gives the value of the spring constant (k), confirming the validity of Hooke’s Law.
By conducting this experiment, one can observe that the extension of the spring is proportional to the applied force, thereby verifying Hooke’s Law.
When a disc rolls down an inclined plane, it has two types of kinetic energy: translational and rotational. The translational kinetic energy is due to the movement of the center of mass of the disc along the incline, while the rotational kinetic energy is due to the disc spinning around its axis asRead more
When a disc rolls down an inclined plane, it has two types of kinetic energy: translational and rotational. The translational kinetic energy is due to the movement of the center of mass of the disc along the incline, while the rotational kinetic energy is due to the disc spinning around its axis as it rolls.
For the solid disc, its moment of inertia plays an important role in determining how the energy is distributed between these two types of kinetic energy. For the disc that rolls without slipping, there exists a relationship between its linear velocity and its angular velocity, connecting the translational motion with the rotational motion.
When the total kinetic energy of the disc is analyzed, it is evident that the rotational kinetic energy is a part of the total energy. Once the contributions of both translational and rotational kinetic energy are evaluated, it is found that the ratio of the rotational kinetic energy to the total kinetic energy of a rolling disc is one to three. This means that for every part of kinetic energy contributed by rotation, three parts are from translation, thereby showing the balance between these two forms of energy in rolling motion.
When a body is projected from the ground at an angle to the horizontal, its angular momentum about the position of projection changes as it moves through its entire flight. At first, the body has angular momentum, depending on its speed and the distance between the point of projection and its centerRead more
When a body is projected from the ground at an angle to the horizontal, its angular momentum about the position of projection changes as it moves through its entire flight. At first, the body has angular momentum, depending on its speed and the distance between the point of projection and its center of mass, respectively. However, as the body continues through its trajectory, gravitational forces are applied. The weight of the body creates a torque about the point of projection, which affects its angular momentum.
As the body rises, the vertical component of its velocity decreases due to gravity, and it eventually reaches its maximum height before descending. During this time, the torque exerted by gravity continuously alters the angular momentum. Since the gravitational force always acts vertically downward through the centre of mass, the distance between line of action of gravitational force and pivot changes due to motion. Therefore, a loss of angular momentum occurs due to rotation at the original axis. All the angular momentum is reduced during this whole course of motion in the projected body due to both gravitational torque as well as displacement of the moving body with the pivot point.
Define the term strain. Why it has no units and dimensions? What are different types of strain?
Definition of Strain: Strain is a measure of deformation, which gives the displacement between the particles in a material body. Strain is defined as the ratio of change in dimension to the original dimension of the material. In mathematical terms, strain (ε) is expressed as: ε = ΔL / L₀ Here, ΔLRead more
Definition of Strain:
Strain is a measure of deformation, which gives the displacement between the particles in a material body. Strain is defined as the ratio of change in dimension to the original dimension of the material. In mathematical terms, strain (ε) is expressed as:
ε = ΔL / L₀
Here, ΔL represents the change in length and L₀ represents the original length.
Why Strain Has No Units and Dimensions:
Strain is a dimensionless quantity because it is a ratio of two lengths (change in length to original length), which means that both numerator and denominator have the same units. Hence, the units cancel out, and one gets a unitless value. Since it is dimensionless, therefore it has no dimensions in the context of physical measurement.
Different Types of Strain:
1. Tensile Strain: It takes place when a material is stretched. It is defined as the change in length divided by the original length (elongation).
2. Compressive Strain: It takes place when a material is compressed. It is defined as the change in length divided by the original length (shortening).
3. Shear Strain: It is that strain which arises due to shear forces applied on a material. It is defined as the change in angle between two lines originally at right angles, expressed as the ratio of lateral displacement to the original length.
4. Volumetric Strain: It is the change in volume per unit original volume of a material, occurring under uniform pressure.
These types of strain are important for understanding how materials respond to various forces and deformations.
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Define the term stress. Give its units and dimensions. Describe the different types of stress.
Definition of Stress: Stress is defined as the force exerted per unit area within materials. It results from external forces applied to an object, resulting in deformation. Mathematically, stress is represented as σ = F / A where σ is the stress F is the applied force and A is the cross-sectional arRead more
Definition of Stress:
Stress is defined as the force exerted per unit area within materials. It results from external forces applied to an object, resulting in deformation. Mathematically, stress is represented as
σ = F / A
where σ is the stress F is the applied force and A is the cross-sectional area over which the force is applied.
Units:
The SI unit of stress is Pascal (Pa), which is equivalent to Newton per square meter (N/m²).
Dimensions:
The dimensions of stress are represented as [M L⁻² T⁻²], where M is mass, L is length, and T is time.
Different Types of Stress:
1. Tensile Stress: This takes place when any material undergoes stretching or pull forces. The force per unit area along the direction of an applied force defines it.
2. Compressive Stress: It arises when compressive forces compress or squeeze the material. Force per unit area in a direction opposite to an applied force is defined by compressive stress.
3. Shear Stress: It happens when a material is subjected to forces that cause one layer of the material to slide over another. It is the force applied parallel to the surface divided by the area of the surface.
4. Volumetric Stress: This stress is due to changes in volume because of uniform pressure applied in all directions. It is significant in fluid mechanics and materials science.
Understanding these different types of stress is important in analyzing material behavior under various loading conditions.
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Explain elastic behaviour of solids on the basis of mechanical spring-ball model of a solid.
Elasticity of Solids: Elastic behavior refers to the phenomenon of an ability to regain the original configuration shape and size when the applied external force is removed. This is critical in understanding how a material deforms and recovers under stress. The mechanical spring-ball model is a usefRead more
Elasticity of Solids:
Elastic behavior refers to the phenomenon of an ability to regain the original configuration shape and size when the applied external force is removed. This is critical in understanding how a material deforms and recovers under stress. The mechanical spring-ball model is a useful representation to explain this concept.
Mechanical Spring-Ball Model:
1. Basic Concept: In a mechanical spring-ball model, atoms or molecules in a solid can be visualized as balls held together by springs. The springs are equivalent to the interatomic forces or bonding forces between the atoms.
2. Elastic Deformation: When external force is applied to the solid, balls go a little away from their equilibrium positions; as a result springs stretch or compress because of which elastic deformation takes place. In this type of deformation, the shape of the material changes, but the material remains intact.
3. Restoring Forces: When the applied force is withdrawn, the springs exert restoring forces that restore the balls to their original positions. This is because the interatomic forces are elastic in nature; the material can return to its original shape and size.
4. Elastic Limit: Elastic behavior is seen up to a certain limit called the elastic limit. If the applied force exceeds this limit, then the springs might get permanently deformed and the deformation will be plastic. In this case, the solid cannot regain its original shape.
5. Mathematical Representation: This elastic region has a relationship that can be stated using Hooke’s Law that relates stress as proportional to strain (σ = Eε, where E is the modulus of elasticity).
In this mechanical model of a spring-ball, interatomic forces and arrangements of atoms come into play about deformation and recovery characteristics of the solids under stress.
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Define the terms deforming force, elasticity and plasticity. What are perfectly elastic and perfectly plastic bodies? Give examples.
Deforming Force: A deforming force is any force applied on the outside to change the shape or size of a material. Such forces can result in either elastic or plastic deformation depending on the intensity of the applied force and material characteristics. Deforming forces can occur from tension, comRead more
Deforming Force: A deforming force is any force applied on the outside to change the shape or size of a material. Such forces can result in either elastic or plastic deformation depending on the intensity of the applied force and material characteristics. Deforming forces can occur from tension, compression, shear, or torsion.
Elasticity:
Elasticity refers to a material’s ability to regain its original shape and size after removal of the applied deforming force. Some materials exhibit this so that they can be deformable in a temporary manner while characteristically, stress and strain are directly proportional within the elastic limit. Restoration to the original state is made possible by interatomic forces.
Plasticity:
Plasticity is the ability of a material to undergo permanent deformation once it has been subjected to a deforming force greater than its elastic limit. Once the force is removed, the material will not return to its original shape and will have altered its structure. Plastic behavior is characterized by a lack of proportionality between stress and strain once the elastic limit has been reached.
Perfectly Elastic Bodies:
Perfectly elastic bodies are materials that can return to their original shape and size after the removal of any applied deforming force, regardless of the magnitude of the force, as long as it does not exceed the material’s elastic limit. They exhibit linear stress-strain behavior and follow Hooke’s Law throughout their entire range of deformation.
Example: An ideal rubber band behaves like a perfectly elastic body, which means it stretches and returns to its original shape once the deforming force is removed.
Perfectly Plastic Bodies:
Perfectly plastic bodies are the ones that fail to regain the shape once they lose the externally applied deforming force. Thus, they go for permanent deformation with no evidence of elastic deformation. After yield point, more stress will create only plastic deformation with no further rise in the level of stress.
Example: Modeling clay is an example of a perfectly plastic body, as it can be easily shaped and will retain the new shape after the applied force is removed.
Understanding these concepts is essential for studying material behavior under various forces and applications in engineering and materials science.
See lessDefine the term compressibility. Give its units and dimensions.
Definition of Compressibility: Compressibility is a measure of the ability of a substance to decrease in volume under pressure. It quantifies how much a material will compress when subjected to an applied external force. The compressibility (β) of a substance is defined as the fractional change in vRead more
Definition of Compressibility:
Compressibility is a measure of the ability of a substance to decrease in volume under pressure. It quantifies how much a material will compress when subjected to an applied external force. The compressibility (β) of a substance is defined as the fractional change in volume per unit increase in pressure.
Mathematical Expression:
Compressibility is mathematically expressed as:
β = – (1/V) * (ΔV/ΔP)
where:
– β is the compressibility,
– V is the initial volume,
– ΔV is the change in volume,
– ΔP is the change in pressure.
Units:
The SI unit of compressibility is the reciprocal of pressure which is usually expressed in terms of:
– Pa⁻¹ (Pascal inverse) or
– N⁻¹ m² (Newton inverse per square meter).
Dimensions:
The dimensions for compressibility can be written as:
[M⁻¹ L³ T²]
Where:
M is mass,
L is length,
T is time.
In various disciplines such as fluid mechanics, material science, and engineering, it is highly relevant to know the compressibility because gases and liquids often react to varying conditions of pressure.
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Define bulk modulus of elasticity. Give its units and dimensions.
Definition of Bulk Modulus of Elasticity: The bulk modulus of elasticity is the coefficient of a medium's resistance toward uniform compression, defined as a ratio of relative change in volume to the intensity of pressure by which the material volume is decreased or increased. For mathematical expreRead more
Definition of Bulk Modulus of Elasticity:
The bulk modulus of elasticity is the coefficient of a medium’s resistance toward uniform compression, defined as a ratio of relative change in volume to the intensity of pressure by which the material volume is decreased or increased. For mathematical expression the following is employed:
K=-ΔP/(ΔV\V)
where K is the bulk modulus ΔP is the change in pressure ΔV is the change in volume and V is the original volume.
Units:
The SI unit of bulk modulus is Pascal (Pa), which is equal to Newton per square meter (N/m²).
Dimensions:
The dimensions of bulk modulus are expressed as [M L⁻¹ T⁻²], where M is mass L is length and T is time.
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Define modulus of elasticity. Give its units and dimensions. What are different types of moduli of elasticity?
Definition of Modulus of Elasticity: Modulus of elasticity is a measure of the ability of a material to deform elastically under the influence of a force. It is a measure of the ratio of stress (force per unit area) to strain (deformation) in a material. The modulus indicates how much a material wilRead more
Definition of Modulus of Elasticity:
Modulus of elasticity is a measure of the ability of a material to deform elastically under the influence of a force. It is a measure of the ratio of stress (force per unit area) to strain (deformation) in a material. The modulus indicates how much a material will deform under a given load.
Units:
The SI unit of modulus of elasticity is Pascal (Pa), which is equal to Newton per square meter (N/m²).
Dimensions:
The modulus of elasticity is expressed in units of [M L⁻¹ T⁻²], where M stands for mass, L for length, and T for time.
Some Common Moduli of Elasticity:
1. Young’s Modulus (E): It is the tensile or compressive elasticity of a material, that is, the ratio of tensile stress to tensile strain.
2. Bulk Modulus (K): It represents the resistance offered by a material to uniform compression. It is defined as the ratio of the change in pressure to the relative decrease in volume.
3. Shear Modulus (G): Also known as the modulus of rigidity, it measures a material’s response to shear stress. It is defined as the ratio of shear stress to shear strain.
4. Poisson’s Ratio (ν): It is a measure of the ratio of transverse strain to axial strain in a material subjected to axial stress, but not a modulus in the strict sense.
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State Hooke’s law. How can it be verified experimentally?
Hooke’s Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, provided the elastic limit is not exceeded. Mathematically, it is expressed as: F = -kx where F is the restoring force exerted by the spring k is the spring constant andRead more
Hooke’s Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, provided the elastic limit is not exceeded. Mathematically, it is expressed as:
F = -kx
where F is the restoring force exerted by the spring k is the spring constant and x is the displacement from the equilibrium position.
Experimental Verification:
To verify Hooke’s Law experimentally, the following steps can be followed:
1. Setup: Attach a spring vertically to a fixed support and hang weights from the free end of the spring. Use a ruler or measuring tape to measure the displacement.
2. Apply Weights: Gradually add known weights to the spring and measure the extension (displacement) of the spring each time a weight is added. Ensure that the weights are added incrementally and the spring is not overstretched.
3. Record Data: Record the weight applied (force) and the corresponding displacement of the spring.
4. Plotting Graph: Plot a graph of the applied force (y-axis) against the displacement (x-axis). According to Hooke’s Law, the graph should be a straight line passing through the origin, indicating that the force is directly proportional to the displacement.
5. Determine the Spring Constant: The slope of the linear graph gives the value of the spring constant (k), confirming the validity of Hooke’s Law.
By conducting this experiment, one can observe that the extension of the spring is proportional to the applied force, thereby verifying Hooke’s Law.
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A disc is rolling on the inclined plane. What is the ratio of its rotation KE to the total KE?
When a disc rolls down an inclined plane, it has two types of kinetic energy: translational and rotational. The translational kinetic energy is due to the movement of the center of mass of the disc along the incline, while the rotational kinetic energy is due to the disc spinning around its axis asRead more
When a disc rolls down an inclined plane, it has two types of kinetic energy: translational and rotational. The translational kinetic energy is due to the movement of the center of mass of the disc along the incline, while the rotational kinetic energy is due to the disc spinning around its axis as it rolls.
For the solid disc, its moment of inertia plays an important role in determining how the energy is distributed between these two types of kinetic energy. For the disc that rolls without slipping, there exists a relationship between its linear velocity and its angular velocity, connecting the translational motion with the rotational motion.
When the total kinetic energy of the disc is analyzed, it is evident that the rotational kinetic energy is a part of the total energy. Once the contributions of both translational and rotational kinetic energy are evaluated, it is found that the ratio of the rotational kinetic energy to the total kinetic energy of a rolling disc is one to three. This means that for every part of kinetic energy contributed by rotation, three parts are from translation, thereby showing the balance between these two forms of energy in rolling motion.
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See lessA body is projected from the ground with some angle to the horizontal. What happens to the angular momentum about the initial position in this motion?
When a body is projected from the ground at an angle to the horizontal, its angular momentum about the position of projection changes as it moves through its entire flight. At first, the body has angular momentum, depending on its speed and the distance between the point of projection and its centerRead more
When a body is projected from the ground at an angle to the horizontal, its angular momentum about the position of projection changes as it moves through its entire flight. At first, the body has angular momentum, depending on its speed and the distance between the point of projection and its center of mass, respectively. However, as the body continues through its trajectory, gravitational forces are applied. The weight of the body creates a torque about the point of projection, which affects its angular momentum.
As the body rises, the vertical component of its velocity decreases due to gravity, and it eventually reaches its maximum height before descending. During this time, the torque exerted by gravity continuously alters the angular momentum. Since the gravitational force always acts vertically downward through the centre of mass, the distance between line of action of gravitational force and pivot changes due to motion. Therefore, a loss of angular momentum occurs due to rotation at the original axis. All the angular momentum is reduced during this whole course of motion in the projected body due to both gravitational torque as well as displacement of the moving body with the pivot point.
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