Using differential equation of linear S.H.M, obtain the expression for (a) velocity in S.H.M., (b) acceleration in S.H.M.
Answers
Answer:
Explanation:
Given:
- Differential Equation of linear S.H.M
To Find:
- To obtain an expression for velocity in S.H.M
- To obtain an expression for acceleration in S.H.M
Solution:
The differential equation for S.H.M is given by,
where ω² = k/m , k is the force constant and m is the mass of the particle.
We know that acceleration is the change in velocity of the body, and velocity is the change in displacement that is,
Therefore,
where x = displacement of the particle
We know,
This can be written as,
But we know that the acceleration of a particle in SHM is,
a = -ω² x
Therefore,
By integration on both sides we get,
Now we know that,
Therefore,
When the particle reaches the extreme position, ie, when x = A, the velocity becomes 0.
Hence,
Substitute the value of C,
v² = -ω²x² + ω²A²
v² = ω² (A² - x²)
Dɪғғᴇʀᴇɴᴛɪᴀʟ ғᴏʀᴍ ᴏғ Lɪɴᴇᴀʀ S.H.M :
Wʜᴇʀᴇ,
- , is the angular frequency of the particle performing the linear simple harmonic motion.
Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,
In any S.H.M, there will always be acting as a restoring force which will try to bring the object back to the mean position. This force will result in an acceleration in the object.
Tʜᴇʀᴇғᴏʀᴇ,
[NOTE ➛ -ve sign indicates that the force will be opposite to the displacement.]
Wʜᴇʀᴇ,
- F be the restoring force.
- x be the displacement of the object from the mean position.
- K is the force per unit displacement.
☆ According to Newton's second law of motion,
➛ Putting the value of F from the equation (i), we get
➛ From the differential form of the linear S.H.M,
- ω =
- ➾ ω² =
➛ Putting the above value in the equation, we get
Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,
Wʜᴇʀᴇ,
- is the change in velocity w.r.t time.
➛ Thus, from equation (a)
Wʜᴇʀᴇ,
- is the change in displacement w.r.t time, i.e. velocity (v).
↝ Let us integrates this equation as,
↝ In the above equation, when displacement is maximum,
- x = A (amplitude)
↝ And at extreme point,
- v = 0
➛
➛ C =
↝ Let us substitute this in the equation as,