To solve this problem, we need to break down the motion of the slotted link and how it affects the rod sliding along the curved channel. We will calculate the radial and circumferential components of both velocity and acceleration at the specified angle of 45 degrees (which we will convert to radians for our calculations).
Understanding the Geometry and Motion
The radius of the channel as a function of the angle θ is given by:
R(θ) = 0.7θ
At θ = 45 degrees, we first convert this angle to radians:
θ = 45° × (π/180) = π/4 radians
Now, substituting θ into the radius function:
R(π/4) = 0.7 × (π/4) ≈ 0.549 m
Calculating the Velocity Components
The angular velocity (ω) is given as 3 rad/s, and the radial distance (R) at θ = π/4 is approximately 0.549 m. The velocity components can be derived as follows:
- Circumferential Velocity (Vθ): This is given by the formula:
Vθ = R × ω
Substituting the values:
Vθ = 0.549 m × 3 rad/s ≈ 1.647 m/s
- Radial Velocity (Vr): Since the radius is changing with time, we need to consider the time derivative of R:
To find the radial velocity, we first need to calculate the derivative of R with respect to time:
R = 0.7θ
Taking the time derivative:
Vr = dR/dt = d(0.7θ)/dt = 0.7(dθ/dt) = 0.7ω
Substituting the angular velocity:
Vr = 0.7 × 3 rad/s = 2.1 m/s
Determining the Acceleration Components
Next, we will calculate the radial and circumferential acceleration components. The angular acceleration (α) is given as 2 rad/s².
- Circumferential Acceleration (Aθ): This is given by:
Aθ = R × α
Substituting the values:
Aθ = 0.549 m × 2 rad/s² ≈ 1.098 m/s²
- Radial Acceleration (Ar): This is composed of two parts: the centripetal acceleration and the tangential acceleration:
Ar = (Vθ²/R) + (R × α)
Calculating the centripetal acceleration:
Vθ²/R = (1.647 m/s)² / 0.549 m ≈ 4.706 m/s²
Now, calculating the tangential acceleration:
R × α = 0.549 m × 2 rad/s² ≈ 1.098 m/s²
Thus, the total radial acceleration is:
Ar = 4.706 m/s² + 1.098 m/s² ≈ 5.804 m/s²
Summary of Results
At θ = 45° (or π/4 radians), we find:
- Circumferential Velocity (Vθ): ≈ 1.647 m/s
- Radial Velocity (Vr): ≈ 2.1 m/s
- Circumferential Acceleration (Aθ): ≈ 1.098 m/s²
- Radial Acceleration (Ar): ≈ 5.804 m/s²
This breakdown provides a comprehensive understanding of the motion of the rod as it slides along the curved channel, influenced by the angular velocity and acceleration of the slotted link.
