Class 11 Exam > Class 11 Questions > A block of mass M slides along a horizontal t...
Start Learning for Free
A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest.
Verified Answer
A block of mass M slides along a horizontal table with speed v0. At x=...
This question is part of UPSC exam. View all Class 11 courses
Most Upvoted Answer
A block of mass M slides along a horizontal table with speed v0. At x=...
The Problem
A block of mass M is sliding along a horizontal table with an initial speed v0. When it reaches x=0, it hits a spring with a constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx, where b is a constant. We need to find the loss in mechanical energy when the block first comes momentarily to rest.
Solution
To find the loss in mechanical energy, we need to calculate the work done by the friction force. The work done by a variable friction force can be found by integrating the product of the force and displacement over the entire path.
Calculating the Work Done
Let's consider a small displacement dx at a distance x from the starting point. The friction force acting on the block can be calculated as f = £mg = bmxg. The work done by this friction force over the small displacement dx is given by dW = f dx.
Since f = bmxg, we have dW = bmxg dx.
Integrating both sides of the equation, we get W = ∫(0 to x) bmxg dx.
Evaluating the integral, we have W = (1/2) bmgx^2.
Calculating the Potential Energy
When the block first comes momentarily to rest, its displacement x is maximum. At this point, all the initial kinetic energy is converted into potential energy stored in the spring.
The potential energy stored in a spring is given by PE = (1/2) kx^2.
Calculating the Loss in Mechanical Energy
The loss in mechanical energy is the difference between the initial kinetic energy and the potential energy stored in the spring.
Initial kinetic energy = (1/2) Mv0^2
Potential energy stored in the spring = (1/2) kx^2
Loss in mechanical energy = (1/2) Mv0^2 - (1/2) kx^2
Substituting the value of x from the equation x = v0^2 / (bg), we have
Loss in mechanical energy = (1/2) Mv0^2 - (1/2) kv0^4 / (b^2 g^2)
Simplifying further, we get
Loss in mechanical energy = (1/2) Mv0^2 (1 - v0^2 / (b^2 g^2))
Conclusion
The loss in mechanical energy when the block first comes momentarily to rest is given by (1/2) Mv0^2 (1 - v0^2 / (b^2 g^2)). This loss occurs due to the work done by the variable friction force.
A block of mass M is sliding along a horizontal table with an initial speed v0. When it reaches x=0, it hits a spring with a constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx, where b is a constant. We need to find the loss in mechanical energy when the block first comes momentarily to rest.
Solution
To find the loss in mechanical energy, we need to calculate the work done by the friction force. The work done by a variable friction force can be found by integrating the product of the force and displacement over the entire path.
Calculating the Work Done
Let's consider a small displacement dx at a distance x from the starting point. The friction force acting on the block can be calculated as f = £mg = bmxg. The work done by this friction force over the small displacement dx is given by dW = f dx.
Since f = bmxg, we have dW = bmxg dx.
Integrating both sides of the equation, we get W = ∫(0 to x) bmxg dx.
Evaluating the integral, we have W = (1/2) bmgx^2.
Calculating the Potential Energy
When the block first comes momentarily to rest, its displacement x is maximum. At this point, all the initial kinetic energy is converted into potential energy stored in the spring.
The potential energy stored in a spring is given by PE = (1/2) kx^2.
Calculating the Loss in Mechanical Energy
The loss in mechanical energy is the difference between the initial kinetic energy and the potential energy stored in the spring.
Initial kinetic energy = (1/2) Mv0^2
Potential energy stored in the spring = (1/2) kx^2
Loss in mechanical energy = (1/2) Mv0^2 - (1/2) kx^2
Substituting the value of x from the equation x = v0^2 / (bg), we have
Loss in mechanical energy = (1/2) Mv0^2 - (1/2) kv0^4 / (b^2 g^2)
Simplifying further, we get
Loss in mechanical energy = (1/2) Mv0^2 (1 - v0^2 / (b^2 g^2))
Conclusion
The loss in mechanical energy when the block first comes momentarily to rest is given by (1/2) Mv0^2 (1 - v0^2 / (b^2 g^2)). This loss occurs due to the work done by the variable friction force.
Attention Class 11 Students!
To make sure you are not studying endlessly, EduRev has designed Class 11 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 11.
|
|
Explore Courses for Class 11 exam
|
|
Similar Class 11 Doubts
Top Courses for Class 11View all
A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest.
Question Description
A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest..
A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest..
Solutions for A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. in English & in Hindi are available as part of our courses for Class 11.
Download more important topics, notes, lectures and mock test series for Class 11 Exam by signing up for free.
Here you can find the meaning of A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. defined & explained in the simplest way possible. Besides giving the explanation of
A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest., a detailed solution for A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. has been provided alongside types of A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. theory, EduRev gives you an
ample number of questions to practice A block of mass M slides along a horizontal table with speed v0. At x=0 hits a spring constant k and begins to experience a friction force. The coefficient of friction is variable and is given by £=bx where b is a constant. Find the loss in mechanical energy when the block has first come momentarily to rest. tests, examples and also practice Class 11 tests.
|
|
Explore Courses for Class 11 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup