Answer:
The dimensionality of B is <em>length</em> per cubic <em>time</em>.
Explanation:
Units for displacement and time are <em>length</em>
and <em>time</em>
, respectively. Then, formula can be tested for dimensional analysis as follows:
![[L] = B\cdot [T]^{3}](https://tex.z-dn.net/?f=%5BL%5D%20%3D%20B%5Ccdot%20%5BT%5D%5E%7B3%7D)
Now, let is clear
to determine its units:
![B = \frac{[L]}{[T]^{3}}](https://tex.z-dn.net/?f=B%20%3D%20%5Cfrac%7B%5BL%5D%7D%7B%5BT%5D%5E%7B3%7D%7D)
The dimensionality of B is <em>length</em> per cubic <em>time</em>.
The total mechanical energy is the sum of the kinetic energy and the gravitational potential energy:
![E=K+U= \frac{1}{2}mv^2 +mgh](https://tex.z-dn.net/?f=E%3DK%2BU%3D%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2%20%2Bmgh)
where m=3.5 kg is Candy's mass, v=1 m/s is her velocity and h=3.5 m is her height. If we replace these numbers, we find the mechanical energy of the system:
Answer:
a) ![T=0.40 N](https://tex.z-dn.net/?f=T%3D0.40%20N)
b) ![T=1.9 s](https://tex.z-dn.net/?f=T%3D1.9%20s)
Explanation:
Let's find the radius of the circumference first. We know that bob follows a circular path of circumference 0.94 m, it means that the perimeter is 0.94 m.
The perimeter of a circunference is:
![P=2\pi r=0.94](https://tex.z-dn.net/?f=P%3D2%5Cpi%20r%3D0.94)
![r=\frac{0.94}{2\pi}=0.15 m](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B0.94%7D%7B2%5Cpi%7D%3D0.15%20m)
Now, we need to find the angle of the pendulum from vertical.
![tan(\alpha)=\frac{r}{L}=\frac{0.15}{0.90}=0.17](https://tex.z-dn.net/?f=tan%28%5Calpha%29%3D%5Cfrac%7Br%7D%7BL%7D%3D%5Cfrac%7B0.15%7D%7B0.90%7D%3D0.17)
![\alpha=9.44 ^{\circ}](https://tex.z-dn.net/?f=%5Calpha%3D9.44%20%5E%7B%5Ccirc%7D)
Let's apply Newton's second law to find the tension.
![\sum F=ma_{c}=m\omega^{2}r](https://tex.z-dn.net/?f=%5Csum%20F%3Dma_%7Bc%7D%3Dm%5Comega%5E%7B2%7Dr)
We use centripetal acceleration here, because we have a circular motion.
The vertical equation of motion will be:
(1)
The horizontal equation of motion will be:
(2)
a) We can find T usinf the equation (1):
![T=\frac {mg}{cos(\alpha)}=\frac{0.04*9.81}{cos(9.44)}=0.40 N](https://tex.z-dn.net/?f=T%3D%5Cfrac%20%7Bmg%7D%7Bcos%28%5Calpha%29%7D%3D%5Cfrac%7B0.04%2A9.81%7D%7Bcos%289.44%29%7D%3D0.40%20N)
We can find the angular velocity (ω) from the equation (2):
![\omega=\sqrt{\frac{Tsin(\alpha)}{mr}}=3.31 rad/s](https://tex.z-dn.net/?f=%5Comega%3D%5Csqrt%7B%5Cfrac%7BTsin%28%5Calpha%29%7D%7Bmr%7D%7D%3D3.31%20rad%2Fs)
b) We know that the period is T=2π/ω, therefore:
![T=\frac{2\pi}{\omega}=\frac{2\pi}{3.31}=1.9 s](https://tex.z-dn.net/?f=T%3D%5Cfrac%7B2%5Cpi%7D%7B%5Comega%7D%3D%5Cfrac%7B2%5Cpi%7D%7B3.31%7D%3D1.9%20s)
I hope it helps you!
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