S.h.m (1 Viewer)

[Ryuu-jin-jakka]

I just don't understand why for b, i did a easy

a)Show that a particle moving in SHM with displacment x = 7cos5t has acceleration given by a = -25 (already proven)

b) Find the times at which the particle will have maximum displacement, and find this maximum displacement

 

[Uncle]

I just don't understand why for b, i did a easy

a)Show that a particle moving in SHM with displacment x = 7cos5t has acceleration given by a = -25 (already proven)

b) Find the times at which the particle will have maximum displacement, and find this maximum displacement

x = 7 cos 5t

Maximum displacement occurs when cos t is at 0, 2π, 4π, 6π .... nπ where n is even.

But you have 5t, so then 5t = nπ, t = nπ/5 where n is an even number from 0 to infinity.

t = 0, 2π/5, 4π/5, 6π/5, 8π/5, 2π

What's cos 0?
One!
But x = 7 cos 0?
Seven!
The maximum displacement will be 7, the maximum amplitude.


You will need to know the properties of the cosine function when finding Fourier series especially when applied to Partial Differential Equations as eigenfunctions of the cosine solution space.

EDIT:

I'll throw in a free graph.
Sorry π isn't marked on the graph, but 2π is approximately equal to 6.28

Remember you are finding maximum displacement, don't worry about what is under the curve.
Most importantly, physically understand the problem, know the terminology.

 

[Ryuu-jin-jakka]

GG ><' lol i fogot that, thx man

 

[Ryuu-jin-jakka]

cool is that part of the La TeX thing ^.^

 

[Drongoski]

I just don't understand why for b, i did a easy

a)Show that a particle moving in SHM with displacment x = 7cos5t has acceleration given by a = -25 (already proven)

b) Find the times at which the particle will have maximum displacement, and find this maximum displacement

x = 7 cos 5t

Max x ( = 7 in magnitude) occurs when cos 5t = 1 (and -1 if considering max as distance from the centre of oscillation, being x=0 in this case)

x = 7 when cos 5t = 1 ==> when 5t = 0 + 2n pi ==> t = 0, 2 pi/5, 4 pi/5 etc

x = -7 when cos 5t = -1 ==> 5t = pi + 2n pi ==> t = pi/5, 3pi/5, pi, etc

 

[Uncle]

cool is that part of the La TeX thing ^.^

Nope.
Software called Maple.
Used in first year maths at UNSW and by engineers, scientists, etc.
Maple can calculate what may take you ages only takes a split second for Maple.

 

[Ryuu-jin-jakka]

The velocity of a particle moving in SHM in a straight lpine is given by v^2 = 4x - x^2 ms-1, where x is displacement in meters

a) Find the two points between which the particle is oscillating

b) Find the center of the motion

c) Find the maximum speed of that particle

d) Find the acceleration of the particle in terms of x

 

[Drongoski]

Edit

Max speed occurs when x = 2 (at centre) =sqrt(n^2 *(a^2 - 0) ) = n*a = 1 x 2 = 2 m/s

particle oscillates between x = 2-2 and x = 2+2, i.e between 0 m and 4m

Acceleration: x(double dot) = -n^2(x-2) = -(x-2) m/s/s