Question

Can someone help me solve these two equations?

Answer 1

Answer:

1a. 0.030 s

1b. 15×10¹⁰ kg m²/s²

Explanation:

Remember order of operations (PEMDAS).

1a. $T_{s} =2\pi \sqrt{\frac{4.5\times10^{-2}\ kg }{2.0\times10^{3}\ kg/s^{2} } }$

Start by dividing the numbers, the powers of 10 (using exponent rules), and the units.

$T_{s} =2\pi \sqrt{\frac{4.5}{2.0} \frac{10^{-2}}{10^{3}} \frac{kg}{kg/s^{2}}}\\T_{s} =2\pi \sqrt{2.25 \times10^{-2-3}\ s^{2}}\\T_{s} =2\pi \sqrt{2.25 \times10^{-5}\ s^{2}}$

Take the square root:

$T_{s} =2\pi \sqrt{2.25} \sqrt{10^{-5}} \sqrt{s^{2}}\\T_{s} =2\pi \times1.5 \times10^{-2.5}\ s$

Finally, multiply (you'll need a calculator):

$T_{s} \approx 0.030\ s$

1b. $K=\frac{1}{2} (6.6\times10^{2}\ kg)(2.11\times10^{4}\ m/s)^{2}$

Start by squaring the last term:

$K=\frac{1}{2} (6.6\times10^{2}\ kg)(2.11^{2}\times10^{4\times2}\ m^{2}/s^{2})\\K=\frac{1}{2} (6.6\times10^{2}\ kg)(4.4521\times10^{8}\ m^{2}/s^{2})$

Now like before, multiply the numbers, the powers of 10 (using exponent rules), and the units:

$K=\frac{1}{2} (6.6\times4.4521)(10^{2+8})(kg\ m^{2}/s^{2})\\K=\frac{1}{2} (29.38386)(10^{10})(kg\ m^{2}/s^{2})\\K\approx15\times10^{10}\ kg\ m^{2}/s^{2}$