Consider a simple pendulum consisting of a small marble of mass 10 g that can be onsidered a point mass suspended from the lower end of an inextensible string of length 80 cm and whose mass is negligible. The upperstring is connected to a fixed point O .The pendulum is shifted by an angle 60º from its vertical position of equilibrium the string being stretched.
a) Taking the level passing through A as reference level. Calculate the potential energy of the pendulum-earth system when the marble is at B. What is the mechanical energy of this system?
b) Mass m is released .Calculate neglecting all frictional forces the velocity of the marble when it passes through A.
Thanks.
and where is A, and B?
O
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| \B
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A
A. The height of B is 80cm(1-cos60) PE is mgh
B. that pe becomes ke.
KE=mgh above
so 1/2 m v^2=mgh
v=sqrt(2gh) where h is the starting position height.
To calculate the potential energy of the pendulum-earth system when the marble is at point B, we need to first calculate the height of point B relative to the reference level. Here's how to do it:
1. First, draw a diagram of the pendulum. Label the points as shown in the question: the fixed point O, the equilibrium position A, and the position B.
2. The length of the string is given as 80 cm. Since the pendulum is shifted by an angle of 60 degrees from its vertical position, the height of point B relative to point A can be calculated using trigonometry.
- The opposite side of the triangle formed is equal to the vertical displacement, which is the vertical distance between A and B.
- The hypotenuse of the triangle formed is equal to the length of the string.
- Using trigonometry, we can find the opposite side by multiplying the hypotenuse by the sine of the angle of 60 degrees.
So, the vertical displacement between A and B is given by:
Vertical displacement = (length of the string) * sin(angle)
Vertical displacement = 0.8 m * sin(60 degrees) (converting 80 cm to meters)
= 0.8 m * √3/2 (using the value of sin(60 degrees) = √3/2)
= 0.8√3/2 m
= 0.4√3 m
3. Now, we can calculate the potential energy at point B. The potential energy in a gravitational field is given by the equation:
Potential Energy = mass * acceleration due to gravity * height
The mass of the marble is given as 10 g, which is equal to 0.01 kg. The acceleration due to gravity is approximately 9.8 m/s^2.
Potential Energy at B = (mass) * (acceleration due to gravity) * (height)
= 0.01 kg * 9.8 m/s^2 * (0.4√3 m)
= 0.0392√3 J
So, the potential energy of the pendulum-earth system when the marble is at point B is approximately 0.0392√3 J.
To calculate the mechanical energy of the system, we need to sum up the potential energy and the kinetic energy at point B:
Mechanical Energy = Potential Energy + Kinetic Energy
Since the marble is at point B, its kinetic energy is zero as it is momentarily at rest before it starts its downward swing.
Therefore, the mechanical energy of the system is equal to the potential energy, which is approximately 0.0392√3 J.
Moving on to part B of the question:
To calculate the velocity of the marble when it passes through point A, we will use the principle of conservation of energy. At point A, the mechanical energy of the system will be conserved, and it will be equal to the sum of potential energy and kinetic energy when the marble is at its maximum height (point B).
Since the marble is at its maximum height at point B, the potential energy at B is equal to the mechanical energy of the system, which we calculated to be approximately 0.0392√3 J.
At point A, all the energy is converted to kinetic energy, as potential energy is zero at its lowest position in the swing.
Therefore:
Mechanical Energy = Potential Energy + Kinetic Energy at point A
0.0392√3 J = 0 + (mass of the marble) * (velocity at point A)^2 * 0.5
0.0392√3 J = 0.01 kg * (velocity at point A)^2 * 0.5
Rearranging the equation to solve for the velocity at point A:
(velocity at point A)^2 = (0.0392√3 J * 2) / (0.01 kg)
(velocity at point A)^2 = 0.0784√3 J / 0.01 kg
(velocity at point A)^2 = 7.84√3 m^2/s^2
Taking square root on both sides:
velocity at point A = √(7.84√3) m/s
velocity at point A ≈ 3.148 m/s
Therefore, neglecting all frictional forces, the velocity of the marble when it passes through point A is approximately 3.148 m/s.