A cart of mass m (including the mass of the crane) has a pendulum hanging from a crane attached to the cart. There is no friction due to the normal force, u = 0, or wheel bearings of the cart wheels. The cart is experiencing and applied force to the left of 10N. The ball at the end of massless rope attached to the crane has a mass mb and the length of the rope tied to the crane and ball is length, L. Express the motion of the ball in terms of m, mb,Fapp (=10N), L and g (acceleration of gravity near the surface of the earth).
Consider tension force on rope. When ball/cart is not moving, T = mbg. If ball swings to an angle theta, O, then the y-component of T is mbg cosO. The cart is also pulling on the crane of the rope with a Fapp that is opposite to the x-component of the T (mbg sinO).
Newton's Second Law says to find the Fnet in both directions, so for very small angles Fnet = Fapp - mbg sinO. Total mass is m + mb, so a (of ball) is nearly equal to (Fapp - mbg sinO)/(m +mb). Once we find a, then we can use kinematics to solve for velocity and position at any given point in time.
To express the motion of the ball in terms of the given variables (m, mb, Fapp, L, and g), we need to analyze the forces acting on the system and apply Newton's second law of motion.
First, let's consider the forces acting on the ball (pendulum). The two main forces acting on the ball are its weight (mg) and the tension in the rope (T). The tension in the rope provides the centripetal force required to keep the ball moving in a circular path.
The magnitude of the tension in the rope can be calculated using the equation:
T = mb * a, ... (Equation 1)
where a is the acceleration of the ball.
The acceleration of the ball (a) in terms of angular velocity (ω) and the length of the rope (L) is given by:
a = ω^2 * L, ... (Equation 2)
where ω is the angular velocity of the ball.
Now, let's analyze the forces acting on the cart. The only force acting on the cart is the applied force (Fapp) to the left.
Since there is no friction due to the normal force or wheel bearings, the net force acting on the cart is:
Fnet = Fapp ... (Equation 3)
According to Newton's second law, the net force is equal to the mass of the system (m) multiplied by the acceleration of the system (ac).
Fnet = m * ac ... (Equation 4)
Since the ball is connected to the cart by a massless rope, the acceleration of the ball is the same as the acceleration of the cart (ac).
Now, we can equate Equations 3 and 4:
Fapp = m * ac.
Substituting Equation 1 into this equation and rearranging gives:
Fapp = m * (mb * a).
Substituting Equation 2 for a gives:
Fapp = m * (mb * ω^2 * L).
Now, solving for ω, we get:
ω^2 = Fapp / (m * mb * L).
Taking the square root, we have:
ω = √(Fapp / (m * mb * L)).
Finally, substituting Equation 2 for ω:
a = (√(Fapp / (m * mb * L)))^2 * L.
Simplifying and rearranging, we get the expression for the motion of the ball:
a = Fapp * L / (m * mb).