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exis [7]
3 years ago
11

A vertical spring with spring constant 23.15 N/m is hanging from a ceiling. A small object is attached to the lower end of the s

pring, and the spring stretches to its equilibrium length. The objects is then pulled down a distance of 19.79 cm and released. The speed of the object at a distance 7.417 from the equilibrium point is 0.7286 m/s. What is the mass of the object?
Physics
1 answer:
umka2103 [35]3 years ago
8 0

Answer:

m= 1.47 kg

Explanation:

Given:

spring constant, K = 23.15 N/m

Displacement, x= 19.79 cm = 0.1979 m

at, x₁= 7.417 cm, v₁= 0.7286 m/s

Now,

x = 19.79 cos ( ωt)

on substituting the values, we get

7.417 x 10⁻² = 19.79 x 10⁻² cos (ωt)

or

cos(ωt) = 0.374

or

ωt = 67.98°

also

v = -0.1979×ωsin (ωt)

also

\omega=\frac{2\pi}{T}

on substituting the values in the above equation, we get

0.7286 = -0.1979 \frac{2\pi}{T} sin ( 67.98°)

3.68 =-\frac{2\pi}{T}(0.927)

or

T = 1.582 sec

also,

T=2\pi\sqrt{\frac{m}{k}}

where, m is the mass

on substituting the values, we have

1.582=2\pi\sqrt{\frac{m}{23.15}}

on squaring both sides and solving, we have

m= 1.47 kg

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First we need to define an Frame of reference. Lets put the x axis unit vector \hat{i} pointing east,  with the y axis unit vector \hat{j} pointing south, so the positive angle is south of east. For this, we got for the first force:

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\vec{F}_3  =  83.7 \ N \ \hat{j}  + 59.9 \ N \ \hat{i}

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But, we need the magnitude and the direction.

To find the magnitude, we can use the Pythagorean theorem.

|\vec{R}| = \sqrt{R_x^2 + R_y^2}

|\vec{F}_3| = \sqrt{(59.9 \ N)^2 + (83.7 \ N)^2}

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this is the magnitude.

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\theta = arctan(\frac{F_{3_y}}{F_{3_x}})

\theta = arctan(\frac{83.7 \ N }{ 59.9 \ N })

\theta = 57 \° 24 ' 48''

and this is the angle south of east.

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