Question

Collar P slides outward at a constant relative speed along rod AB, which rotates counterclockwise with a constant angular velocity of 3.65 rad/s. Knowing r = 14.34 in when theta = 0, and the collar reaches B when theta = 98 degrees, determine the magnitude of the acceleration of the collar P just as it reaches B in ft/s^2.

Answer 1

The magnitude of the acceleration of collar P just as it reaches B is 31 in/s^2.

Step-by-step explanation:

Given: r = 14.34 in when θ = 0, θ = 98 degrees at B, and the angular velocity of the rod, AB, is 3.65 rad/s.

To find the magnitude of the acceleration of collar P, we need to determine its position, angular displacement, and time.

Using the given information, we can find that the arc length, s, covered by collar P is:

s = r * θ = (14.34 in) * (98 degrees * (π/180)) = 26.01 in

Now, we need to determine the time it takes for collar P to reach B. Since the angular velocity is constant, we can use the formula:

ω = Δθ/Δt

Δt = Δθ/ω = (98 degrees * (π/180)) / 3.65 rad/s = 1.69 s

The magnitude of the acceleration of collar P can then be determined using the formula:

a = (v_f - v_i) / Δt = (0 - v_i) / Δt = -v_i / Δt

Since collar P is sliding outward, its initial velocity is 0. We can calculate it using:

v_i = ω * r = 3.65 rad/s * 14.34 in = 52.41 in/s

Substituting the values, we get:

a = -v_i / Δt = -(52.41 in/s) / 1.69 s = -31 in/s^2

Therefore, the magnitude of the acceleration of collar P just as it reaches B is 31 in/s^2.

Answer 2

a= 23.65 ft/s²

Step-by-step explanation:

given

r= 14.34m

ω=3.65rad/s

Ф=Ф₀ + ωt

t = Ф - Ф₀/ω

= (98-0)×
`(\pi)/(180)`/3.65

98°= 1.71042 rad

1.7104/3.65

t= 0.47 s

r₁(not given)

assuming r₁ =20 in

r₁ = r₀ + ut(uniform motion)

u = r₁ - r₀/t

r₀ = 14.34 in= 1.195 ft

r₁ = 20 in = 1.67 ft

= (1.667 - 1.195)/0.47

0.472/0.47

u= 1.00ft/s

acceleration at collar p

a=rω²

= 1.67 × 3.65²

a = 22.25ft/s²

acceleration of collar p related to the rod = 0

coriolis acceleration = 2ωu

= 2× 3.65×1 = 7.3 ft/s²

acceleration of collar p

= 22.5j + 0 + 7.3i

√(22.5² + 7.3²)

the magnitude of the acceleration of the collar P just as it reaches B in ft/s²

a= 23.65 ft/s²