
A ball is suspended by a thread of length . It is pulled a side until the thread makes an angle of with a vertical. How much work is done against gravity? The ball is now released. Find the velocity at the lowest point. Ignore air resistance and take .
Answer
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Hint: When a ball is suspended by a thread, it acts like a pendulum. When the particle is at a mean position, its potential energy is zero, kinetic energy is maximum , but kinetic energy is zero. Work done by a ball is due to work done by gravity which is given by potential energy.
Use the work energy theorem, according to which change in kinetic energy is equal to work done and work done is equal to potential energy. It means change in potential energy is equal to change in kinetic energy.
And we can calculate velocity at the lowest point.
Formula used: Work done due to gravity
By work energy theorem
Here is a mean position.
is an extreme position.
.
Complete step by step solution
We have a blue suspended by a thread. is the mean position and is the extreme position.
Mass of ball is given by this is the length of thread.
=
Since
Here, thread is at
Height through which ball is raised,
Work done due to gravity is given by
By work energy theorem,
According to this, the work done by all forces acting on a particle equals change in the K.E of the particle.
Hence work done by ball is
And velocity at lowest point is
Note
Work energy theorem: The work done by the net force on a particle is equal to the change in the particle's kinetic energy.
The work energy theorem can be derived from Newton's second law: kinetic energy is directly proportional to the mass of the object and to the square of its velocity.
Potential energy is energy an object has because of its position relative to the same other object.
The formula of potential energy depends on the force acting on two objects.
Use the work energy theorem, according to which change in kinetic energy is equal to work done and work done is equal to potential energy. It means change in potential energy is equal to change in kinetic energy.
And we can calculate velocity at the lowest point.
Formula used: Work done due to gravity
By work energy theorem
Here
Complete step by step solution
We have a blue suspended by a thread.

Mass of ball is given by
Since
Here, thread is at
Height through which ball is raised,
Work done due to gravity is given by
By work energy theorem,
According to this, the work done by all forces acting on a particle equals change in the K.E of the particle.
Hence work done by ball is
And velocity at lowest point is
Note
Work energy theorem: The work done by the net force on a particle is equal to the change in the particle's kinetic energy.
The work energy theorem can be derived from Newton's second law: kinetic energy is directly proportional to the mass of the object and to the square of its velocity.
Potential energy is energy an object has because of its position relative to the same other object.
The formula of potential energy depends on the force acting on two objects.
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