# Rod OA rotates about O in a horizontal plane. The motion of the 0.5-lb collar B is defined by the relations r = 10 + 6 cos πt and θ = π(4t2 – 8t), where r is expressed in inches, t in seconds, and θ in radians. Determine the radial and transverse components of the force exerted on the collar when (a) t = 0, (b) t = 0.5 s. Fig. P12.66

Question-AnswerCategory: Engineering MechanicsRod OA rotates about O in a horizontal plane. The motion of the 0.5-lb collar B is defined by the relations r = 10 + 6 cos πt and θ = π(4t2 – 8t), where r is expressed in inches, t in seconds, and θ in radians. Determine the radial and transverse components of the force exerted on the collar when (a) t = 0, (b) t = 0.5 s. Fig. P12.66
GME asked 1 year ago

Rod OA rotates about O in a horizontal plane. The motion of the 0.5-lb collar B is defined by the relations r = 10 + 6 cos πt and θ = π(4t2 – 8t), where r is expressed in inches, t in seconds, and θ in radians. Determine the radial and transverse components of the force exerted on the collar when (a) t = 0, (b) t = 0.5 s.
Fig. P12.66

Step: 1

Calculate the mass of the collar B .

Here, weight of the collar B is W, and acceleration due to gravity is g.
Substitute 0.5 lb for W, and  for g.

Hence, the mass of the collar B  is

Step: 2

Let r and  be the polar coordinates of the collar B. Express the radial component of the collar B (r).
…… (1)
Express the transverse component of the collar B .
…… (2)
Calculate the derivatives of the radial component of the collar B with respect to time t.
Differentiate Equation (1) with respect to time t.

But derivative of r with respect to time (t) is . Therefore,
…… (3)

Step: 3

Differentiate Equation (3) with respect to time t.

But derivative of  with respect to time (t) is .
…… (4)
Calculate the derivatives of the transverse component of the collar B with respect to time t.
Differentiate Equation (2) with respect to time t.

But derivative of  with respect to time (t) is .
…… (5)
Differentiate Equation (5) with respect to time t.

But derivative of  with respect to time (t) is . Therefore,
…… (6)

Step: 4

Calculate the radial and transverse components of the force exerted on the collar at .
Substitute 0 for t in Equation (1).

Substitute 0 for t in Equation (3).

Substitute 0 for t in Equation (4).

Step: 5

Substitute 0 for t in Equation (2).

Substitute 0 for t in Equation (5).

Express the radial component of the acceleration of the collar B .
…… (7)
Substitute  for , 16 in. for r, and  for .

Step: 6

Express the transverse component of the acceleration of the collar B.
…… (8)
Substitute 16 in. for r,  for , 0 for , and  for .

Express the radial component of the force exerted on the collar B.
…… (9)
Substitute for m, and  for .

Therefore, the radial component of the force exerted on the collar B is.

Step: 7

Express the transverse component of the force exerted on the collar B.
…… (10)
Substitute for m, and  for .

Therefore, the transverse component of the force exerted on the collar B is.
Use Equations (1) and (2) and calculate the radial and angular co-ordinate of collar at 0 s.
Substitute 0 s in Equation (1) to calculate the collar position.

Substitute 0 s in Equation (2) to calculate the collar position.

Therefore, the collar radial co-ordinate at 0 s is 16 in., while the angular co-ordinate is.

Step: 8

Calculate the radial and transverse components of the force exerted on the collar at .
Substitute 0.5 for t in Equation (1).

Substitute 0.5 for t in Equation (3).

Substitute 0.5 for t in Equation (4).

Step: 9

Substitute 0.5 for t in Equation (2).

Substitute 0.5 for t in Equation (5).

Substitute 0 for , 10 in. for r, and  for .

Step: 10

Substitute 10 in. for r,  for  for , and  for .

Express the radial component of the force exerted on the collar B.

Substitute for m, and  for .

Hence, the radial component of the force exerted on the collar B is.
Express the transverse component of the force exerted on the collar B.

Substitute for m, and  for .

Hence, the transverse component of the force exerted on the collar B is.
Use Equations (1) and (2) and calculate the radial and angular co-ordinate of collar at .
Substitute 0.5 s in Equation (1) to calculate the collar position.

Substitute 0 s in Equation (2) to calculate the collar position.

Angular co-ordinate of  indicates that the rod has rotated by one complete revolution. Therefore,

Therefore, the collar radial co-ordinate is 10 in., while the angular co-ordinate is.