A cylinder with a circular hole is rolling without slipping on a fixed curved surface as shown. The cylinder would have a weight of 16 lb without the hole, but with the hole it has a weight of 15 lb. Knowing that at the instant shown the disk has an angular velocity of 5 rad/s clockwise, determine (a) the angular acceleration of the disk, (b) the components of the reaction force between the cylinder and the ground at this instant.
Fig. P16.161
(a)
The angular acceleration of the disk.
Answer to Problem 16.161RP
The angular acceleration of the disk is
Explanation of Solution
Given information:
The weight of the cylinder without hole is
The weight of the cylinder with hole is
The angular velocity of the disk is
Calculation:
Consider the acceleration due to gravity as
Consider that the mass center of the cylinder is G and it lies at a distance b from center A, and C is the contact point between cylinder and the curved surface is the origin of the coordinate system.
Consider that the radius of cylinder is r and the radius of the curved surface is R.
Sketch the geometry of the cylinder as shown in Figure 1.
Refer to Figure 1.
Calculate the position vector
The position vector of P with respect to C.
The position vector of G with respect to C.
The position vector of A with respect to C.
The cylinder rolls without slipping on a fixed curved surface hence, the horizontal component of acceleration of point C is
Calculate the acceleration of point C
Substitute 0 for
The acceleration of the disk at A is
Calculate the acceleration of point A
Substitute
Calculate the acceleration of point P
Here,
Substitute
Calculate the acceleration of point G
Substitute
Calculate the acceleration of point A
Substitute
Subtract Equation (2) from Equation (1) as shown below.
Substitute
Calculate the velocity of point A
Calculate the vertical component of acceleration of point A
Substitute
Calculate the acceleration of point G
Substitute
Calculate the effective force at the mass center
Sketch the Free Body Diagram of the cylinder as shown in Figure 2.
Refer to Figure 2.
Apply the Equilibrium of moment about C as shown below.
Hence, the angular acceleration of the disk is
(b)
The components of the reaction force between the cylinder and the ground.
Answer to Problem 16.161RP
The component of the reaction along x direction is
The component of the reaction along y direction is
Explanation of Solution
Given information:
The weight of the cylinder without hole is
The weight of the cylinder with hole is
The angular velocity of the disk is
Calculation:
Refer to part (a).
The angular acceleration of the disk is
Refer to Figure 2.
Apply the Equilibrium of force along x direction as shown below.
Substitute 0 for
Hence, the component of the reaction along x direction is
Apply the Equilibrium of force along y direction as shown below.
Sketch the cylinder with hole as shown in Figure 3.
Refer to Figure 3.
Calculate the area of solid cylinder
Calculate the center of gravity of solid cylinder in vertical direction
Calculate the area of hole
Calculate the center of gravity of hole in vertical direction
Calculate the distance
Substitute
Calculate the component of the reaction along y direction as shown below.
Substitute
Therefore, the component of the reaction along y direction is
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