Answer :
Answer:
(a) v = 1.54m/s, (b) Θ = 80.6° and (c) E = 52.92J
Explanation:
(a) At the lowest point, the total energy is giving by:
[tex] E_{T}_{0} = E_{K}_{0} + E_{P}_{0} = \frac {1}{2} mv_{0}^{2} + mgy_{0} [/tex]
where,[tex] E_{K} [/tex]: kinetic energy,[tex] E_{P} [/tex]: potential energy, m: pendulum's mass, v: speed of the pendulum, g:gravity and y: heigh of the pendulum.
[tex] E_{T}_{0} = \frac {1}{2} mv_{0}^{2} + mg(0) = \frac {1}{2} (1.5)(8.4)^{2} = 52.92 J [/tex] (1)
When the string is at 64° from the vertical, the total energy is:
[tex] E_{T}_{f} = \frac {1}{2} mv_{f}^{2} + mgy_{f} [/tex] (2)
At this point, [tex] y_{f} [/tex] is:
[tex] y_{f} = L - Lcos(\Theta) [/tex] (3)
where L: longitude of the pendulum
[tex] y_{f} = 4.3 m (1 -cos(64)) = 2.42 m [/tex]
By conservation of energy, we can calculate the speed of the string, at 64 ° to the vertical. Equaling equations (1) and (2):
[tex] E_{T}_{0} = E_{T}_{f} = \frac {1}{2} mv_{f}^{2} + mgy_{f} [/tex]
[tex] v_{f} = \sqrt \frac{2(E_{T}_{0} - mgy_{f})}{mg}} [/tex]
[tex] v_{f} = \sqrt \frac{2(52.92 - 1.5 \cdot 9.8 \cdot 2.42)}{1.5 \cdot 9.8}} = 1.54 \frac{m}{s} [/tex]
(b) Smilarly, by conservation of energy we can find the greatest angle, assuming that vf = 0 at the greatest angle reached:
[tex] \frac {1}{2} mv_{0}^{2} + mgy_{0} = \frac {1}{2} mv_{f}^{2} + mgy_{f} [/tex]
[tex] y_{f} = \frac {E_{0}}{mg} = \frac {52.92}{(1.5)(9.8)} = 3.6 m [/tex]
Using the heigh calculated in equation (3) we can find the angle:
[tex] \Theta = Arccos (\frac {L - y_{f}}{L}) = Arccos (\frac {4.3 - 3.6}{4.3}) = 80.6^\circ [/tex]
(c) The total mechanical energy at the lowest point is giving by:
[tex] E_{T} = E_{K} + E_{P} = \frac {1}{2} mv_{0}^{2} + 0 = \frac {1}{2} (1.5)(8.4)^{2} = 52.92 J [/tex]
Have a nice day!
For a pendulum swinging stone, the lowest position is the zero level or reference position.
- (a) The speed when the string is at 64 ˚ to the vertical is 3.6 meters.
- (b) The greatest angle with the vertical that the string will reach during the stone's motion is 80.6 degrees.
- (c)The total mechanical energy of the system is 52.92 joules.
What is pendulum?
Pendulum is the body which is pivoted a point and perform back and forth motion around that point by swinging due to the influence of gravity.
The mass of the pendulum is 1.5 kg.
The length of the string is 4.3 meters.
The speed of the stone is 8.4 m/s.
- (a) The speed when the string is at 64 ˚ to the vertical-
The initial kinetic energy is equal to the final kinetic energy,
[tex]\dfrac{1}{2}mv_o^2+mgy_o=\dfrac{1}{2}mv_f^2+mgy_f[/tex]
Here, (m) is mass, (v) is velocity, (y) is the position and (subscript o and f) used for initial and final.
Put the values as,
[tex]\dfrac{1}{2}(1.5)(8.4)^2+(1.5)(9.81)(0)=\dfrac{1}{2}(1.5)(v_f)^2+(1.5)(9.81)(4.3-4.3\times\cos(64))\\[/tex]
Solve the above equation, we get,
[tex]v_f=1.54\rm m/s[/tex]
- (b) The greatest angle with the vertical that the string will reach during the stone's motion-
When the final velocity becomes zero, then the string will reach greatest height during stone's motion. Thus put [tex]v_f[/tex] equal to zero in above equation,
[tex]\dfrac{1}{2}(1.5)(8.4)^2+(1.5)(9.81)(0)=\dfrac{1}{2}(1.5)(0)^2+(1.5)(9.81)(y_f)\\[/tex]
Solve it further, we get,
[tex]y_f=3.6\rm m[/tex]
Thus the greatest angle is,
[tex]y_f=L-L\cos\theta\\3.6=4.3-4.3(\cos\theta)\\\theta=80.6^o[/tex]
The greatest angle with the vertical that the string will reach during the stone's motion is 80.6 degrees.
- (c)The total mechanical energy of the system-
Total mechanical energy is the sum of kinetic energy and potential energy. Thus,
[tex]E_T=\dfrac{1}{2}(1.5)(8.4)^2+(1.5)(9.81)(0)\\E_T=52.92 \rm J[/tex]
Thus the total mechanical energy of the system is 52.92 joules
.
- (a) The speed when the string is at 64 ˚ to the vertical is 3.6 meters.
- (b) The greatest angle with the vertical that the string will reach during the stone's motion is 80.6 degrees.
- (c)The total mechanical energy of the system is 52.92 joules.
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