Concept explainers
Determine the impulse exerted on the plate of Prob. 18.31 during the impact by (a) the obstruction at B, (b) the support at A.
18.31 A square plate of side a and mass m supported by a ball-and-socket joint at A is rotating about the y axis with a constant angular velocity ω = ω0j when an obstruction is suddenly introduced at B in the xy plane. Assuming the impact at B to be perfectly plastic (e = 0), determine immediately after the impact (a) the angular velocity of the plate, (b) the velocity of its mass center G.
Fig. P18.31
(a)
The impulse exerted
Answer to Problem 18.32P
The impulse exerted
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:
Show the diagram of the system as in Figure (1).
The length of the diagonal of a square is obtained by multiplying the side with
The expression for the angular velocity in the
Here,
The expression for the angular velocity in the
The unit vectors along the
The expression for the initial angular momentum about the mass center
Here,
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
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
The matrix multiplication for vector product is done.
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
Show the forces acting on the plate as in Figure (2).
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
Substitute
Therefore, the impulse exerted
(b)
Find the impulse exerted
Answer to Problem 18.32P
The impulse exerted
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:
Calculate the velocity along the x, y and z axes
Consider the Equation (3).
Substitute 0 for
Substitute
The expression for the linear momentum of the system as follows:
Substitute 0 for
Therefore, the impulse exerted
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Chapter 18 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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