To find the resultant acceleration of the particle on the given path, we need to analyze the motion of the rigid body and apply the principles of calculus and physics. The path of the particle is defined by the equation \( y = a \log_e \sec(x/a) \), and we know that the angular speed of the tangent to the curve is \( 2 \) rad/s. Let's break this down step by step.
Step 1: Understanding the Given Equation
The equation \( y = a \log_e \sec(x/a) \) describes the trajectory of the particle. Here, \( a \) is given as \( 1 \) m, so we can substitute that into our equation:
Thus, the equation simplifies to:
y = log_e sec(x)
Step 2: Finding the Derivative
To find the acceleration, we first need to determine the velocity of the particle. The velocity can be found by differentiating the position function with respect to time. We can use the chain rule for this purpose:
Let \( x(t) = a t \) since the particle moves along the x-axis. Then, we can express \( y \) in terms of \( t \):
y(t) = log_e sec(at)
Now, differentiate \( y \) with respect to \( t \):
Using the chain rule:
- dy/dt = (dy/dx) * (dx/dt)
We first find dy/dx:
dy/dx = d(log_e sec(x))/dx = sec(x) tan(x)
Now, since dx/dt = a (where a = 1 m), we have:
dy/dt = sec(x) tan(x) * 1 = sec(x) tan(x)
Step 3: Finding the Acceleration
Acceleration is the derivative of velocity. Therefore, we need to differentiate \( dy/dt \) with respect to time:
To find d²y/dt², we apply the product rule:
d²y/dt² = d/dt(sec(x) tan(x))
Using the chain rule, we have:
- d²y/dt² = (d(sec(x))/dx)(dx/dt)tan(x) + sec(x)(d(tan(x))/dx)(dx/dt)
Now, we need to evaluate this at \( x = \frac{\pi}{4} \) m:
At \( x = \frac{\pi}{4} \):
- sec(x) = sec(π/4) = √2
- tan(x) = tan(π/4) = 1
Substituting these values into our acceleration equation gives:
d²y/dt² = (√2)(1)(1) + (√2)(1)(1) = 2√2 m/s²
Step 4: Resultant Acceleration
The resultant acceleration of the particle is the vector sum of the tangential and centripetal accelerations. The tangential acceleration is given by the angular speed multiplied by the radius (which is \( a \) in this case), and the centripetal acceleration is given by \( a \cdot \omega^2 \), where \( \omega \) is the angular speed.
Given that \( \omega = 2 \) rad/s:
- Tangential acceleration = a * angular speed = 1 * 2 = 2 m/s²
- Centripetal acceleration = a * ω² = 1 * (2)² = 4 m/s²
Now, we can find the resultant acceleration using the Pythagorean theorem:
Resultant acceleration = √(tangential² + centripetal²) = √(2² + 4²) = √(4 + 16) = √20 = 2√5 m/s²
Final Result
Thus, the resultant acceleration of the particle when \( x = \frac{\pi}{4} \) m is:
2√5 m/s²
