(Mass does not affect the pendulum's swing. The longer the length of string, the farther the pendulum falls; and therefore, the longer the period, or back and forth swing of the pendulum. The greater the amplitude, or angle, the farther the pendulum falls; and therefore, the longer the period.)
Answer:
![E_r(6)=4.35614\ MPa](https://tex.z-dn.net/?f=E_r%286%29%3D4.35614%5C%20MPa)
Explanation:
= Strain = 0.49
= 3.1 MPa
At t = Time = 32 s
= 0.41 MPa
= Time-independent constant
Stress relation with time
![\sigma=\sigma _0exp\left(-\frac{t}{\tau}\right)](https://tex.z-dn.net/?f=%5Csigma%3D%5Csigma%20_0exp%5Cleft%28-%5Cfrac%7Bt%7D%7B%5Ctau%7D%5Cright%29)
at t = 32 s
![0.41=3.1exp\left(-\frac{32}{\tau}\right)\\\Rightarrow exp\left(-\frac{32}{\tau}\right)=\frac{0.41}{3}\\\Rightarrow -\frac{32}{\tau}=ln\frac{0.41}{3}\\\Rightarrow \tau=-\frac{32}{ln\frac{0.41}{3}}\\\Rightarrow \tau=16.0787\ s](https://tex.z-dn.net/?f=0.41%3D3.1exp%5Cleft%28-%5Cfrac%7B32%7D%7B%5Ctau%7D%5Cright%29%5C%5C%5CRightarrow%20exp%5Cleft%28-%5Cfrac%7B32%7D%7B%5Ctau%7D%5Cright%29%3D%5Cfrac%7B0.41%7D%7B3%7D%5C%5C%5CRightarrow%20-%5Cfrac%7B32%7D%7B%5Ctau%7D%3Dln%5Cfrac%7B0.41%7D%7B3%7D%5C%5C%5CRightarrow%20%5Ctau%3D-%5Cfrac%7B32%7D%7Bln%5Cfrac%7B0.41%7D%7B3%7D%7D%5C%5C%5CRightarrow%20%5Ctau%3D16.0787%5C%20s)
The time independent constant is 16.0787 s
![E_{r}(t)=\frac{\sigma(t)}{\epsilon_0}](https://tex.z-dn.net/?f=E_%7Br%7D%28t%29%3D%5Cfrac%7B%5Csigma%28t%29%7D%7B%5Cepsilon_0%7D)
At t = 6
![\\\Rightarrow E_{r}(6)=\frac{\sigma(6)}{\epsilon_0}](https://tex.z-dn.net/?f=%5C%5C%5CRightarrow%20E_%7Br%7D%286%29%3D%5Cfrac%7B%5Csigma%286%29%7D%7B%5Cepsilon_0%7D)
From the first equation
![\sigma(t)=\sigma _0exp\left(-\frac{t}{\tau}\right)\\\Rightarrow \sigma(6)=3.1exp\left(-\frac{6}{16.0787}\right)\\\Rightarrow \sigma(6)=2.13451](https://tex.z-dn.net/?f=%5Csigma%28t%29%3D%5Csigma%20_0exp%5Cleft%28-%5Cfrac%7Bt%7D%7B%5Ctau%7D%5Cright%29%5C%5C%5CRightarrow%20%5Csigma%286%29%3D3.1exp%5Cleft%28-%5Cfrac%7B6%7D%7B16.0787%7D%5Cright%29%5C%5C%5CRightarrow%20%5Csigma%286%29%3D2.13451)
![E_r(6)=\frac{2.13451}{0.49}\\\Rightarrow E_r(6)=4.35614\ MPa](https://tex.z-dn.net/?f=E_r%286%29%3D%5Cfrac%7B2.13451%7D%7B0.49%7D%5C%5C%5CRightarrow%20E_r%286%29%3D4.35614%5C%20MPa)
![E_r(6)=4.35614\ MPa](https://tex.z-dn.net/?f=E_r%286%29%3D4.35614%5C%20MPa)
This problem is about the rate of the current. It's important to know that refers to the quotient between the electric charge and the time, that's the current rate.
![I=\frac{Q}{t}](https://tex.z-dn.net/?f=I%3D%5Cfrac%7BQ%7D%7Bt%7D)
Where Q = 2.0×10^−4 C and t = 2.0×10^−6 s. Let's use these values to find I.
![\begin{gathered} I=\frac{2.0\times10^{-4}C}{2.0\times10^{-6}\sec } \\ I=1.0\times10^{-4-(-6)}A \\ I=1.0\times10^{-4+6}A \\ I=1.0\times10^2A \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20I%3D%5Cfrac%7B2.0%5Ctimes10%5E%7B-4%7DC%7D%7B2.0%5Ctimes10%5E%7B-6%7D%5Csec%20%7D%20%5C%5C%20I%3D1.0%5Ctimes10%5E%7B-4-%28-6%29%7DA%20%5C%5C%20I%3D1.0%5Ctimes10%5E%7B-4%2B6%7DA%20%5C%5C%20I%3D1.0%5Ctimes10%5E2A%20%5Cend%7Bgathered%7D)
<em>As you can observe above, the division of the powers was solved by just subtracting their exponents.</em>
<em />
<h2>Therefore, the rate of the current flow is 1.0×10^2 A.</h2>
Answer:
Explanation:
Force is the change in momentum over time
F = Δp/Δt
1. Calculate the change in momentum
p₁ = mv₁ = 1000 kg × 10 m/s = 10 000 kg·m·s⁻¹
p₂ = 0
Δp = p₂ - p₁= (0 - 10 000) kg·m·s⁻¹ = -10 000 kg·m·s⁻¹
2. Calculate the force
![\begin{array}{rcl}F & = & \dfrac{\Delta p}{\Delta t}\\\\& = & \dfrac{-10 000 \text{ kg$\cdot$m$\cdot$ s}^{-1}}{\text{ 0.5 s}}\\\\& = & \textbf{-20 000 N}\\\end{array}\\\text{The negative sign shows that the force is exerted opposite to the direction of motion.}\\\text{The magnitude of the force is $\large \boxed{\textbf{20 000 N}}$}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brcl%7DF%20%26%20%3D%20%26%20%5Cdfrac%7B%5CDelta%20p%7D%7B%5CDelta%20t%7D%5C%5C%5C%5C%26%20%3D%20%26%20%5Cdfrac%7B-10%20000%20%5Ctext%7B%20kg%24%5Ccdot%24m%24%5Ccdot%24%20s%7D%5E%7B-1%7D%7D%7B%5Ctext%7B%200.5%20s%7D%7D%5C%5C%5C%5C%26%20%3D%20%26%20%5Ctextbf%7B-20%20000%20N%7D%5C%5C%5Cend%7Barray%7D%5C%5C%5Ctext%7BThe%20negative%20sign%20shows%20that%20the%20force%20is%20exerted%20opposite%20to%20the%20direction%20of%20motion.%7D%5C%5C%5Ctext%7BThe%20magnitude%20of%20the%20force%20is%20%24%5Clarge%20%5Cboxed%7B%5Ctextbf%7B20%20000%20N%7D%7D%24%7D)