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A solid sphere of mass 2 kg and a massless string of total length 4 meters which is wrapped around the equator of the sphere (total length includes the length of wrapped string and the length of the string hanging from the ceiling). The sphere then falls vertically while rolling down (without slipping) the whole wrapped string and falls on top of an inclined plane. Due to the collision with the inclined plane, it loses all its kinetic energy within a very short time and starts rolling along the inclined plane with zero initial velocity. Consider that a total of 10 Joules of energy is lost during the motion of the sphere along the planes. The magnitude of gravitational acceleration g = 9.8 ๐/๐ 2.
a. Find the value of the moment of inertia of the sphere about its axis of rotation.
b. What is the magnitude of angular velocity of the sphere when it touches the inclined plane?
c. How far along the second inclined plane (45ยบ) on the right will the center of the sphere travel before it comes to a stop?
A target in a shooting board consists of a vertical square wooden board, 0.250 m on a side and with a mass 750 g and pivots around a horizontal axis along its top edge. The board is stuck face-on at its center by a bullet of mass 1.90 g, travelling at 360 m/s that remains embedded in the board.
(a) (2 marks) What is the angular speed of the board just after the bulletโs impact?
(b) (2 marks) What maximum height does the center of the board reach from the equilibrium before it starts swinging down again?
(c) (2 marks) What minimum bullet speed is needed for the board to swing all the way over after the impact?
A target in a shooting board consists of a vertical square wooden board, 0.250 m on a side and with a mass 0.750 g and pivots around a horizontal axis along its top edge. The board is stuck face-on at its center by a bullet of mass 1.90 g, travelling at 360 m/s that remains embedded in the board.
a) (2 marks) What is the angular speed of the board just after the bulletโs impact?
(b) (2 marks) What maximum height does the center of the board reach from the equilibrium before it starts swinging down again?
(c) (2 marks) What minimum bullet speed is needed for the board to swing all the way over after the impact?
x * (dy)/(dx) = x ^ 2 + 5y
3 d^ 2 y dx^ 2 + dy dx -14y=0;y(0)=1,y^ prime (0)= -1
ย A target in a shooting board consists of a vertical square wooden board, 0.250 m on a side and with a mass 0.750 g and pivots around a horizontal axis along its top edge. The board is stuck face-on at its center by a bullet of mass 1.90 g, travelling at 360 m/s that remains embedded in the board.
(a) (2 marks) What is the angular speed of the board just after the bulletโs impact?
(b) (2 marks) What maximum height does the center of the board reach from the equilibrium before it starts swinging down again?
(c) (2 marks) What minimum bullet speed is needed for the board to swing all the way over after the impact?
- A pinball machine launch ramp consisting of a spring and a 30 degree ramp of length ๐ฟ as shown in Fig. 1.
(a) (3 marks) If the spring is compressed a distance ๐ฅ from its equilibrium position and is then released at ๐ก = 0, the pinball (a sphere of mass ๐ and radius ๐) reaches the top of the ramp at ๐ก = ๐. Derive the expression for the spring constant ๐ in terms of ๐, ๐, ๐ฅ, and ๐ฟ. [Assume that the friction is sufficient, and the ball begins rolling without slipping immediately after launch.]
(b) (2 marks) What is the potential energy of the ball when it is at the midpoint of the ramp?
(c) (3 marks) Derive the expression of the speed of the ball immediately after being launched in terms of ๐ and ๐ฟ.
The pattern of a standing wave consists of 5 loops. If the length of the string is L, then the distance between two consecutive node-antinode is:
A juggler tosses three balls alternately a vertically upward. Each ball has an initial velocity of 5 m/s. How high does each ball rise? How long does each ball remain in the air?
A bomb is dropped from a plane which is flying with a constant horizontal velocity. Locate the bomb after 1 second in relation to the plane.
