Concept explainers
The kinetic energy (U) lost when plate hits the obstruction at B.
Answer to Problem 18.52P
The kinetic energy (U) lost when plate hits the obstruction at B is
Explanation of Solution
Given information:
The mass of the square plate is m.
The side of a square plate is a.
The angular velocity
Assume the impact to be perfectly plastic that is
Calculation:
Draw the diagram of the system as in Figure (1).
The length of the diagonal of a square is obtained by multiplying the side with
Write the expression for the angular velocity in the
Here,
Write the expression for the angular velocity in the
The unit vectors along the
Write the expression for the initial angular momentum about the mass center
Here,
Write the expression for the moment of inertia in the
Here, m is the mass and a is the side of the square plate.
Due to symmetry, moment of inertia in the
Write the expression for the angular momentum about
Substitute
Calculate the angular velocity at B
Here,
Substitute
The matrix multiplication for vector product is done.
The corner B does not rebound after impact. Therefore, the velocity of B after impact in the
Calculate the angular velocity about the mass center
Here,
Substitute
Substitute 0 for
Write the matrix multiplication for vector product is done.
Write the expression for the angular momentum about A as follows:
Here,
Calculate the angular momentum about G using the formula:
Substitute 0 for
Substitute
The matrix multiplication is done for vector product.
The initial velocity of mass center (
Calculate the initial momentum about A using the relation:
Here,
Substitute
Draw the forces acting on the plate as in Figure (2).
Write the expression for the moment about A as follows:
The matrix multiplication for vector product is done.
Substitute Equation (6) and Equation (7).
Equate
Equate
Equate
Calculate the velocity along the x, y and z axes
Substitute 0 for
Substitute
Calculate the kinetic energy of the system before impact
Substitute 0 for
Calculate the kinetic energy of the system after impact
Substitute
Calculate the loss in kinetic energy (U) using the formula:
Substitute
Thus, the kinetic energy (U) lost when plate hits the obstruction at B is
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Chapter 18 Solutions
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