Question

# A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 8.50 N is applied. A 0.530-kg particle rests on a frictionless h

Other
A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 8.50 N is applied. A 0.530-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x = 5.00 cm and released from rest at t = 0. (Assume that the direction of the initial displacement is positive.)
(a) What is the force constant of the spring? 280 N/m
(b) What are the angular frequency (?), the frequency, and the period of the motion?
f = 3.6817 Hz
T = 0.27161 s
(c) What is the total energy of the system? 0.35 J
(d) What is the amplitude of the motion? 5 cm
(e) What are the maximum velocity and the maximum acceleration of the particle?
$$\displaystyle{v}_{{\max}}={1.1561}\frac{{m}}{{s}}$$
$$\displaystyle{a}_{{\max}}={26.73}\frac{{m}}{{s}^{{{2}}}}$$
(f) Determine the displacement x of the particle from the equilibrium position at t = 0.500 s.
(g) Determine the velocity and acceleration of the particle when t = 0.500 s. (Indicate the direction with the sign of your answer.)
v = _________________ $$\displaystyle\frac{{m}}{{s}}$$
a = _________________ $$\displaystyle\frac{{m}}{{s}^{{{2}}}}$$

2020-11-15

a)$$\displaystyle{k}={\frac{{{F}}}{{{x}}}}$$=283.3N/m
b)$$\displaystyle{w}=\sqrt{{{\frac{{{k}}}{{{m}}}}}}$$==23.121rad/s
$$\displaystyle{f}={\frac{{{w}}}{{{2}\pi}}}$$=3.68Hz
$$\displaystyle{T}={\frac{{{1}}}{{{f}}}}$$=0.2717s
c)$$\displaystyle{E}={0.5}{k}{A}^{{{2}}}$$=0.354J
d)5cm
e) $$\displaystyle{v}_{{\max}}={A}{w}={1.1561}\frac{{m}}{{s}}$$
$$\displaystyle{a}_{{\max}}={A}{w}^{{{2}}}={26.73}\frac{{m}}{{s}^{{{2}}}}$$
f) $$\displaystyle{x}={A}{\cos{{\left({w}{t}\right)}}}={5}{\cos{{\left({23.121}\cdot{0.5}\right)}}}={2.677}{c}{m}$$
g) $$\displaystyle{v}=-{A}{w}{\sin{{\left({w}{t}\right)}}}={0.9765}\frac{{m}}{{s}}$$
$$\displaystyle{a}=-{A}{w}^{{{2}}}{\cos{{\left({w}{t}\right)}}}={14.31}\frac{{m}}{{s}^{{{2}}}}$$
a) F=kx.
$$\displaystyle{8.50}={k}{\left({3}\cdot{10}^{{-{2}}}\right)}$$
$$\displaystyle{k}={283.33}{N}{m}^{{-{1}}}$$
b) $$\displaystyle{W}^{{{2}}}={\frac{{{k}}}{{{m}}}}$$
$$\displaystyle{W}^{{{2}}}={\frac{{{283}}}{{{0.530}}}}$$
$$\displaystyle{W}={23.16067977}{r}{a}\frac{{d}}{{s}}$$
c) $$\displaystyle{W}={\frac{{{2}\pi}}{{{T}}}}$$
$$\displaystyle{T}={\frac{{{\left\lbrace{W}\right\rbrace}{\left\lbrace{2}\pi\right\rbrace}}}{}}$$
$$\displaystyle{T}={\frac{{{23.16067977}}}{{{2}\pi}}}$$
$$\displaystyle{T}={3.678812717}{s}.$$
Frequency $$\displaystyle={\frac{{{1}}}{{{p}{e}{r}{i}{o}{d}}}}$$
$$\displaystyle={\frac{{{1}}}{{{3}}}}{.678812717}$$
$$\displaystyle={.272}{H}{z}$$
d) $$\displaystyle{A}={5.00}{c}{m}.$$
e) $$\displaystyle{V}=-{A}{W}{\sin{{W}}}{t}$$
max velocity is when t=1.779406359 s
$$\displaystyle{A}={5}\cdot{10}^{{-{2}}}{m}$$
$$\displaystyle{W}={23.16067977}$$
$$\displaystyle{V}=-{\left({5}\cdot{10}^{{-{2}}}\right)}{\left({22.16067977}\right)}{\sin{{\left({23.160679}\ldots\right)}}}$$
$$\displaystyle{V}=-{1.02}{m}{s}^{{-{1}}}$$
$$\displaystyle{a}=-{A}{W}{\left\lbrace^{2}\right\rbrace}{\cos{{W}}}{t}$$
max acceleration is when t =0
$$\displaystyle{a}=-{A}{W}^{{{2}}}{\cos{{W}}}{t}$$
$$\displaystyle{a}=-{\left({5}\cdot{10}^{{-{2}}}\right)}{\left({22.16067977}\right)}^{{{2}}}{\cos{{\left({22.1606}\ldots\right.}}}$$
$$\displaystyle{a}=-{24.2}{m}{s}^{{-{2}}}$$
f) $$\displaystyle{x}={A}{\cos{{W}}}{t}$$
$$\displaystyle{x}={\left({5}\cdot{10}^{{-{2}}}\right)}{\left({\cos{{\left({22.16067977}\right)}}}\right)}{\left({0.500}\right)}$$
x=0.9185806285 m
g) $$\displaystyle{v}=-{A}{w}{\sin{{\left({w}{t}\right)}}}={0.9765}\frac{{m}}{{s}}$$
$$\displaystyle{a}=-{A}{w}^{{{2}}}{\cos{{\left({w}{t}\right)}}}={14.31}\frac{{m}}{{s}^{{2}}}$$