How can I show that the sphere of radius R performs a simple harmonic movement. how can i set its reference point and make the f
ree body diagram.
I have the torque sum equation which is equal to the moment of inertia by angular acceleration
I have the torque sum equation which is equal to the moment of inertia by angular acceleration
1 answer:
Explanation:
Draw a free body diagram of the pendulum (the combination of the sphere and the massless rod). There are three forces on the pendulum:
Weight force mg at the center of the sphere,
Reaction force in the x direction at the pivot,
Reaction force in the y direction at the pivot.
Sum the torques about the pivot O.
∑τ = I d²θ/dt²
mg (L sin θ) = I d²θ/dt²
For small θ, sin θ ≈ θ.
mg L θ = I d²θ/dt²
Since d²θ/dt² is directly proportional to θ, this fits the definition of simple harmonic motion.
If you wish, you can use parallel axis theorem to find the moment of inertia about O:
I = Icm + md²
I = ⅖ mr² + mL²
mg L θ = (⅖ mr² + mL²) d²θ/dt²
gL θ = (⅖ r² + L²) d²θ/dt²
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Answer:
20 J
Explanation:
Given:
Weight of the book is,
Height or displacement of the book is,
The work done on the book to raise it to a height of 2 m on a shelf is against gravity. The gravitational force acting on the book is equal to its weight. Now, in order to raise it, an equal amount of force must be applied in the opposite direction.
So, the force applied by me should be equal to weight of the body and in the upward direction. The displacement is also in the upward direction.
Now, work done by the applied force is equal to the product of force applied and displacement of book in the direction of the applied force.
Therefore, work done is given as:
Therefore, the work done to raise a book to a height 2 m from the floor is 20 J.