A pendulum is suspended inside a car vertically suddenly the car is given a horizontal acceleration 'a'.The maximm angle through which the pendulum deflects is ?answer is 2tan inverse 'a'divided by g(acceleration due to gravity please solve this i have given lot of time but not having any clue to solve this.
The pendulum has velocity at arctana/g
To solve this problem, we can use the concept of simple harmonic motion and realize that the pendulum will behave like a harmonic oscillator. Here's the step-by-step solution:
Step 1: Determine the forces acting on the pendulum when the car is accelerated horizontally.
When the car accelerates horizontally, two forces act on the pendulum: the tension in the string (T) and the component of the gravitational force (mg) that acts tangentially to the pendulum's path. The net force can be determined by considering the forces in the horizontal and vertical directions separately.
In the horizontal direction, the net force is responsible for accelerating the pendulum. The tension in the string does not provide any horizontal force component, so the only horizontal force is the component of the gravitational force.
F_horizontal = mg*sin(theta) ---(Equation 1)
In the vertical direction, the net force must be equal to the centripetal force required to keep the pendulum moving in a circular path.
F_vertical = T - mg*cos(theta) = m * a ---(Equation 2)
Step 2: Equate the horizontal and vertical forces.
Equating the equations from step 1, we have:
mg*sin(theta) = m * a
Step 3: Simplify the equation.
Divide both sides of the equation by m and rearrange:
g*sin(theta) = a
Step 4: Solve for the maximum angle of deflection.
To find the maximum angle of deflection, we need to find the value of theta that satisfies the equation. However, the equation is nonlinear, so solving it analytically is complex.
Instead, we can make a small-angle approximation, which assumes that the maximum angle of deflection is small (less than 10 degrees). Under this assumption, we can approximate sin(theta) as theta in radians.
Therefore, we can rewrite the equation as:
g * theta = a
Simplifying further:
theta = a/g
Step 5: Convert the result to the desired form.
The given answer is 2*tan^(-1)(a/g). To achieve this, we can rewrite "theta = a/g" as:
theta = tan^(-1)(a/g)
Then multiply both sides by 2:
2 * theta = 2 * tan^(-1)(a/g)
Therefore, the maximum angle through which the pendulum deflects is 2 * tan^(-1)(a/g).
Please note that this solution assumes small-angle approximation and neglects any dampening effects on the system (e.g., air resistance).