The bird perched on the swing shown below has a mass of 42.7 g, and the base of the swing, y = 7.54 cm below the hook, has a mass of 133 g. The swing and bird are originally at rest, and then the bird takes off horizontally at 2.04 m/s. How high will the base of the swing rise above its original level? Disregard friction.
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To calculate the height that the base of the swing will rise above its original level, we need to use the principle of conservation of mechanical energy.
The initial mechanical energy of the system consisting of the swing and the bird is given by:
E_initial = m_swing * g * (y + h) + m_bird * g * h
where:
m_swing = mass of the swing
m_bird = mass of the bird
y = height of the base of the swing below the hook
h = height the base of the swing will rise above its original level
The final mechanical energy of the system is given by:
E_final = m_swing * g * (y + h')
where:
h' = final height of the base of the swing above its original level
Because there is no friction, we can assume that the mechanical energy is conserved, so:
E_initial = E_final
Substituting the values given in the problem, we can write the equation:
m_swing * g * (y + h) + m_bird * g * h = m_swing * g * (y + h')
Now, let's solve for h':
(133 g) * (9.8 m/s^2) * (7.54 cm/100 cm) + (42.7 g) * (9.8 m/s^2) * 0 = (133 g) * (9.8 m/s^2) * (7.54 cm/100 cm) + (42.7 g) * (9.8 m/s^2) * h'
Simplifying the equation:
(133 g) * (9.8 m/s^2) * (7.54 cm/100 cm) = (42.7 g) * (9.8 m/s^2) * h'
(133) * (7.54) = (42.7) * h'
h' = (133 * 7.54) / 42.7
h' = 23.509 cm
Therefore, the base of the swing will rise approximately 23.509 cm above its original level.
To find how high the base of the swing will rise above its original level, we need to use conservation of energy. The initial potential energy of the swing and bird will be equal to the final kinetic energy of the bird when it takes off.
Let's break down the problem step by step:
1. Calculate the initial potential energy (PE_initial) of the swing and bird. The formula for potential energy is PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
PE_initial = (mass of swing + mass of bird) * g * h_initial
= (42.7 g + 133 g) * 9.8 m/s^2 * 0
(since the swing and bird are originally at rest and the height is 0)
PE_initial = 0
2. Calculate the final kinetic energy (KE_final) of the bird when it takes off. The formula for kinetic energy is KE = 0.5 * m * v^2, where m is the mass and v is the velocity.
KE_final = 0.5 * mass of bird * (velocity of bird)^2
= 0.5 * 42.7 g * (2.04 m/s)^2
3. Equate the initial potential energy and final kinetic energy to find the height (h_final) the base of the swing will rise above its original level.
PE_initial = KE_final
0 = 0.5 * 42.7 g * (2.04 m/s)^2
Now let's solve this equation:
0 = 0.5 * 0.0427 kg * (2.04 m/s)^2
0 = 0.5 * 0.0427 kg * 4.1616 m^2/s^2
0 = 0.0427 kg * 2.0808 m^2/s^2
0 = 0.0890 kg * m^2/s^2
Since both sides of the equation are equal to 0, it means that the base of the swing will not rise above its original level.