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The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating?
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The density of a rod of length L varies linearly with position along i...
Density Variation along the Rod
The problem states that the density of the rod varies linearly with position along its length. Let's assume that the density at one end of the rod is ρ and at the other end, it is 3ρ. We can express the density variation as:
ρ(x) = ρ + kx
Where ρ(x) is the density at position x along the rod, ρ is the initial density at one end, x is the position along the rod, and k is the constant representing the rate of density change.
Force Applied to the Rod
Now, let's consider the force applied to the rod. We are given that a force of magnitude f is applied to a point on the rod. To ensure that the rod remains balanced and doesn't rotate, the force must be applied at a specific position along the rod.
Conditions for the Rod to Remain Balanced
For the rod to remain balanced and not rotate, two conditions must be met:
1. The net force acting on the rod must be zero.
2. The torque (rotational force) acting on the rod must be zero.
Net Force Condition
To satisfy the first condition, the net force acting on the rod must be zero. This means that the force applied (f) must be balanced by the weight of the rod at that point.
Torque Condition
To satisfy the second condition, the torque acting on the rod must be zero. Torque is the product of force and the perpendicular distance from the point of application of the force to the axis of rotation.
Calculation of Torque
To calculate the torque, we need to consider the force acting on small elements along the rod. Let's consider a small element of length dx at position x along the rod.
The weight of this element can be calculated as:
dW = (ρ + kx)g dx
Where dW is the weight of the small element, g is the acceleration due to gravity, and dx is the length of the small element.
The force applied to this element is:
dF = f(x) dx
The torque due to this small element is:
dT = (ρ + kx)g dx * (x - x0)
Where x0 is the position at which the force is applied.
Calculation of Total Torque
To calculate the total torque acting on the rod, we need to integrate the torque over the entire length of the rod.
Total torque (T) = ∫ dT
T = ∫ [(ρ + kx)g (x - x0)] dx
T = g ∫ [(ρx + kx^2) - (ρx0 + kx0^2)] dx
T = g [((ρ/2)x^2 + (k/3)x^3) - ((ρ/2)x0^2 + (k/3)x0^3)]
Equating Torque to Zero
To ensure that the rod remains balanced and doesn't rotate, the total torque acting on the rod must be zero. Therefore, we need to solve the equation:
T = g [((ρ/2)x^2 + (k/3)x^3) - ((ρ/2)x0^2 + (k/3)x0^3
The problem states that the density of the rod varies linearly with position along its length. Let's assume that the density at one end of the rod is ρ and at the other end, it is 3ρ. We can express the density variation as:
ρ(x) = ρ + kx
Where ρ(x) is the density at position x along the rod, ρ is the initial density at one end, x is the position along the rod, and k is the constant representing the rate of density change.
Force Applied to the Rod
Now, let's consider the force applied to the rod. We are given that a force of magnitude f is applied to a point on the rod. To ensure that the rod remains balanced and doesn't rotate, the force must be applied at a specific position along the rod.
Conditions for the Rod to Remain Balanced
For the rod to remain balanced and not rotate, two conditions must be met:
1. The net force acting on the rod must be zero.
2. The torque (rotational force) acting on the rod must be zero.
Net Force Condition
To satisfy the first condition, the net force acting on the rod must be zero. This means that the force applied (f) must be balanced by the weight of the rod at that point.
Torque Condition
To satisfy the second condition, the torque acting on the rod must be zero. Torque is the product of force and the perpendicular distance from the point of application of the force to the axis of rotation.
Calculation of Torque
To calculate the torque, we need to consider the force acting on small elements along the rod. Let's consider a small element of length dx at position x along the rod.
The weight of this element can be calculated as:
dW = (ρ + kx)g dx
Where dW is the weight of the small element, g is the acceleration due to gravity, and dx is the length of the small element.
The force applied to this element is:
dF = f(x) dx
The torque due to this small element is:
dT = (ρ + kx)g dx * (x - x0)
Where x0 is the position at which the force is applied.
Calculation of Total Torque
To calculate the total torque acting on the rod, we need to integrate the torque over the entire length of the rod.
Total torque (T) = ∫ dT
T = ∫ [(ρ + kx)g (x - x0)] dx
T = g ∫ [(ρx + kx^2) - (ρx0 + kx0^2)] dx
T = g [((ρ/2)x^2 + (k/3)x^3) - ((ρ/2)x0^2 + (k/3)x0^3)]
Equating Torque to Zero
To ensure that the rod remains balanced and doesn't rotate, the total torque acting on the rod must be zero. Therefore, we need to solve the equation:
T = g [((ρ/2)x^2 + (k/3)x^3) - ((ρ/2)x0^2 + (k/3)x0^3
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The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating?
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The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating?.
The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The density of a rod of length L varies linearly with position along its length such that it increased to thrice its value from one and two other this road is placed on a frictionless horizontal surface and original force of magnitude f is applied to a point on the road at what point should this force be applied such that the road lights on the surface without rotating?.
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