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Luda [366]
3 years ago
14

How to solve arctan[tan(7pi /4)] ...?

Physics
1 answer:
Svet_ta [14]3 years ago
7 0
Well, 
arctan is a bijection from R into (-pi/2 , pi/2)*and 
pi is a period of tangent function: 
so 
as we have : tan(7pi/4) = tan(pi - pi/4) = - tan(pi/4) 
we finally get : 

<span>arctan(tan(7pi/4)) = artan(tan(- pi/4)) = - pi/4 
</span>

I hope my answer has come to your help. Thank you for posting your question here in Brainly. We hope to answer more of your questions and inquiries soon. Have a nice day ahead!
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a 3 kg piece of putty that is moving with a velocity of 10 m/s collides and sticks to an 8 kg bowling ball that was at rest. wha
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The final velocity is 2.7 m/s

Explanation:

We can solve this problem by using the principle of conservation of momentum: in fact, in absence of external forces, the total momentum of the system must be conserved before and after the collision.

Therefore we can write:

p_i = p_f\\m_1 u_1 + m_2 u_2 = (m_1+m_2)v  

where:  

m_1 = 3 kg is the mass of the putty

u_1 = 10 m/s is the initial velocity of the putty (we take its direction as positive direction)

m_2 = 8 kg is the mass of the ball

u_2 = 0 m/s is the initial velocity of the ball (at rest)

v is the final combined velocity of the two putty+ball

Re-arranging the equation and substituting the values, we find the  final combined velocity:

v=\frac{m_1 u_1 + m_2 u_2}{m_1+m_2}=\frac{(3)(10)+0}{3+8}=2.7 m/s

And the positive sign indicates their final direction is the same as the initial direction of the putty.

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A mass is oscillating with amplitude A at the end of a spring.
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A) x=\pm \frac{A}{2\sqrt{2}}

The total energy of the system is equal to the maximum elastic potential energy, that is achieved when the displacement is equal to the amplitude (x=A):

E=\frac{1}{2}kA^2 (1)

where k is the spring constant.

The total energy, which is conserved, at any other point of the motion is the sum of elastic potential energy and kinetic energy:

E=U+K=\frac{1}{2}kx^2+\frac{1}{2}mv^2 (2)

where x is the displacement, m the mass, and v the speed.

We want to know the displacement x at which the elastic potential energy is 1/3 of the kinetic energy:

U=\frac{1}{3}K

Using (2) we can rewrite this as

U=\frac{1}{3}(E-U)=\frac{1}{3}E-\frac{1}{3}U\\U=\frac{E}{4}

And using (1), we find

U=\frac{E}{4}=\frac{\frac{1}{2}kA^2}{4}=\frac{1}{8}kA^2

Substituting U=\frac{1}{2}kx^2 into the last equation, we find the value of x:

\frac{1}{2}kx^2=\frac{1}{8}kA^2\\x=\pm \frac{A}{2\sqrt{2}}

B) x=\pm \frac{3}{\sqrt{10}}A

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