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nignag [31]
3 years ago
9

Solved, just need someone to check! thank you. ​

Physics
1 answer:
Lady bird [3.3K]3 years ago
3 0

Answer:

good job keep it up

Explanation:

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Match each situation below to a letter on the illustration.
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3 years ago
A sphere of mass m" = 2 kg travels with a velocity of magnitude υ") = 8 m/s toward a sphere of mass m- = 3 kg initially at rest,
aleksklad [387]

a) 6.4 m/s

b) 2.1 m

c) 61.6^{\circ}

d) 14.0 N

e) 4.6 m/s

f) 37.9 N

Explanation:

a)

Since the system is isolated (no external forces on it), the total momentum of the system is conserved, so we can write:

p_i = p_f\\m_1 u_1 = m_1 v_1 + m_2 v_2

where:

m_1 = 2 kg is the mass of the 1st sphere

m_2 = 3kg is the mass of the 2nd sphere

u_1 = 8 m/s is the initial velocity of the 1st sphere

v_1 is the final velocity of the 1st sphere

v_2 is the final velocity of the 2nd sphere

Since the collision is elastic, the total kinetic energy is also conserved:

E_i=E_k\\\frac{1}{2}m_1 u_1^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2

Combining the two equations together, we can find the final velocity of the 2nd sphere:

v_2=\frac{2m_1}{m_1+m_2}u_1=\frac{2(2)}{2+3}(8)=6.4 m/s

b)

Now we analyze the 2nd sphere from the moment it starts its motion till the moment it reaches the maximum height.

Since its total mechanical energy is conserved, its initial kinetic energy is entirely converted into gravitational potential energy at the highest point.

So we can write:

KE_i = PE_f

\frac{1}{2}mv^2 = mgh

where

m = 3 kg is the mass of the sphere

v = 6.4 m/s is the initial speed of the sphere

g=9.8 m/s^2 is the acceleration due to gravity

h is the maximum height reached

Solving for h, we find

h=\frac{v^2}{2g}=\frac{(6.4)^2}{2(9.8)}=2.1 m

c)

Here the 2nd sphere is tied to a rope of length

L = 4 m

We know that the maximum height reached by the sphere in its motion is

h = 2.1 m

Calling \theta the angle that the rope makes with the vertical, we can write

h = L-Lcos \theta

Which can be rewritten as

h=L(1-cos \theta)

Solving for \theta, we can find the angle between the rope and the vertical:

cos \theta = 1-\frac{h}{L}=1-\frac{2.1}{4}=0.475\\\theta=cos^{-1}(0.475)=61.6^{\circ}

d)

The motion of the sphere is part of a circular motion. The forces acting along the centripetal direction are:

- The tension in the rope, T, inward

- The component of the weight along the radial direction, mg cos \theta, outward

Their resultant must be equal to the centripetal force, so we can write:

T-mg cos \theta = m\frac{v^2}{r}

where r = L (the radius of the circle is the length of the rope).

However, when the sphere is at the highest point, it is at rest, so

v = 0

Therefore we have

T-mg cos \theta=0

So we can find the tension:

T=mg cos \theta=(3)(9.8)(cos 61.6^{\circ})=14.0 N

e)

We can solve this part by applying again the law of conservation of energy.

In fact, when the sphere is at a height of h = 1 m, it has both kinetic and potential energy. So we can write:

KE_i = KE_f + PE_f\\\frac{1}{2}mv^2 = \frac{1}{2}mv'^2 + mgh'

where:

KE_i is the initial kinetic energy

KE_f is the kinetic energy at 1 m

PE_f is the final potential energy

v = 6.4 m/s is the speed at the bottom

v' is the speed at a height of 1 m

h' = 1 m is the height

m = 3 kg is the mass of the sphere

And solving for v', we find:

v'=\sqrt{v^2-2gh'}=\sqrt{6.4^2-2(9.8)(1)}=4.6 m/s

f)

Again, since the sphere is in circular motion, the equation of the forces along the radial direction is

T-mg cos \theta = m\frac{v^2}{r}

where

T is the tension in the string

mg cos \theta is the component of the weight in the radial direction

m\frac{v^2}{r} is the centripetal force

In this situation we have

v = 4.6 m/s is the speed of the sphere

cos \theta can be rewritten as (see part c)

cos \theta = 1-\frac{h'}{L}

where in this case,

h' = 1 m

L = 4 m

And r=L=4 m is the radius of the circle

Substituting and solving for T, we find:

T=mg cos \theta + m\frac{v^2}{r}=mg(1-\frac{h'}{L})+m\frac{v^2}{L}=\\=(3)(9.8)(1-\frac{1}{4})+(3)\frac{4.6^2}{4}=37.9 N

4 0
3 years ago
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