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
Sazid Ahmad answered 1 year ago
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
.