Problem 3-Conservation of angular momentum and physical pendulum A system is composed of a thin uniform rod of length L= 1 m and mass M = 1 kg, and a bullet of mass mb - 7 g. Initial, the rod is in equilibrium against the force of gravity, with its axis along the vertical direction, pivoted at a distance L/4 from the rod's upper end; the bullet travels along the x-direction, with initial velocity v = 100 m/s, until it collides against the rod at a distance L/4 from the rod's lower end. After the collision, the bullet emerges from the rod with a velocity v' = 50 m/s along the x-direction, and the rod starts rotating around its pivot point. (a) Find the initial angular velocity of the rod. (b) Use conservation of energy to find the maximum angle reached by rod. (c) Find the time it takes for the rod to return to its original position, in the limit where the maximum angle reached by the rod is small (Hint: in part b, use 1 - cos a = 2 (sin a/2)² ).
Problem 3-Conservation of angular momentum and physical pendulum A system is composed of a thin uniform rod of length L= 1 m and mass M = 1 kg, and a bullet of mass mb - 7 g. Initial, the rod is in equilibrium against the force of gravity, with its axis along the vertical direction, pivoted at a distance L/4 from the rod's upper end; the bullet travels along the x-direction, with initial velocity v = 100 m/s, until it collides against the rod at a distance L/4 from the rod's lower end. After the collision, the bullet emerges from the rod with a velocity v' = 50 m/s along the x-direction, and the rod starts rotating around its pivot point. (a) Find the initial angular velocity of the rod. (b) Use conservation of energy to find the maximum angle reached by rod. (c) Find the time it takes for the rod to return to its original position, in the limit where the maximum angle reached by the rod is small (Hint: in part b, use 1 - cos a = 2 (sin a/2)² ).
Problem 3-Conservation of angular momentum and physical pendulum A system is composed of a thin uniform rod of length L= 1 m and mass M = 1 kg, and a bullet of mass mb - 7 g. Initial, the rod is in equilibrium against the force of gravity, with its axis along the vertical direction, pivoted at a distance L/4 from the rod's upper end; the bullet travels along the x-direction, with initial velocity v = 100 m/s, until it collides against the rod at a distance L/4 from the rod's lower end. After the collision, the bullet emerges from the rod with a velocity v' = 50 m/s along the x-direction, and the rod starts rotating around its pivot point. (a) Find the initial angular velocity of the rod. (b) Use conservation of energy to find the maximum angle reached by rod. (c) Find the time it takes for the rod to return to its original position, in the limit where the maximum angle reached by the rod is small (Hint: in part b, use 1 - cos a = 2 (sin a/2)² ).
Can you explain part b
Why is the simplification done like that how did they get omega under angular velocity and where does the 1/2 from the inertia go ?
Transcribed Image Text:(b) Conservation of mechanical energy implice
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Iw² = Mg / (4-cos 8) = Mg ½ sin ² m
2
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Distance O=CM;
4
The frequency of small oscillations of the
physical perdukum is 12 = ( Mg L/4) "
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/where we have used
1-cosα = 2 sin ² x/2
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sin" (ON ²) = = (2) ²
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Transcribed Image Text:VAY, VBx, VBy in the laboratory frame of reference after the
Problem 3-Conservation of angular momentum and physical pendulum
A system is composed of a thin uniform rod of length L = 1 m and mass M = 1 kg, and a bullet of mass mb = 7 g. Initial, the rod is in equilibrium against the
force of gravity, with its axis along the vertical direction, pivoted at a distance L/4 from the rod's upper end; the bullet travels along the x-direction, with
initial velocity v = 100 m/s, until it collides against the rod at a distance L/4 from the rod's lower end. After the collision, the bullet emerges from the rod
with a velocity v' = 50 m/s along the x-direction, and the rod starts rotating around its pivot point. (a) Find the initial angular velocity of the rod. (b) Use
conservation of energy to find the maximum angle reached by rod. (c) Find the time it takes for the rod to return to its original position, in the limit where
the maximum angle reached by the rod is small (Hint: in part b, use 1- cos a = 2 (sin a/2)² ).
Problem 4-Fluids and Heat
A square shaped open water tank of depth Ho is fully filled with water. Suddenly it starts leaking water through a big hole right at the bottom. Assume that
wtional to the square root of
Definition Definition Rate of change of angular displacement. Angular velocity indicates how fast an object is rotating. It is a vector quantity and has both magnitude and direction. The magnitude of angular velocity is represented by the length of the vector and the direction of angular velocity is represented by the right-hand thumb rule. It is generally represented by ω.
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