A pendulum is constructed by attaching a small metal ball to one end of a string of length ?=1.40 m that hangs from the ceiling, as shown in the figure. The ball is released when it is raised high enough for the string to make an angle of ?=20.0∘ with the vertical. With what speed ? is the ball moving at the bottom of its swing? (in m/s) Does the mass of the ball affect the answer? (Choose one) -No, because the ball’s speed is independent of its mass.-Yes, because changing the mass changes the ball's inertia.-No, because the change in string tension counteracts the change in mass.-Yes, because the work done on the ball by the gravitational force depends on the mass. **Understanding Projection Motion**In the diagram, we observe a scenario where a ball is projected horizontally from a certain height.1. **Initial Position and Motion**: - The ball begins at the edge of the platform. - There is an arrow pointing horizontally indicating the initial horizontal velocity.2. **Path of the Ball**: - The trajectory of the ball is depicted as a dotted line curving downward. - This path demonstrates the effect of gravity, which acts on the ball, pulling it downward while it moves forward. 3. **Motion Components**: - **Horizontal Motion**: The green arrow on the right side shows the constant horizontal velocity of the ball. In the absence of air resistance, this velocity remains unchanged. - **Vertical Motion**: The downward curve of the trajectory results from the acceleration due to gravity, which causes the ball to fall toward the ground.This illustration highlights the fundamental principles of projectile motion, where an object is subject to gravity while moving in a horizontal direction. Understanding this concept is crucial in physics as it applies to various real-world scenarios, from sports to engineering.

In a given case, three force are acting on the object : 1) Tension 2) weight(mg) 3) the force that move the high enough for the string Work done against Tension is always zero because T is always perpendicular to displacement. now the potential energy stored by the mass(m) of height L(1-cosθ ) is W = m*g*L(1-cosθ ) ...(1) when the ball moving at the bottom all the potential energy convert in Kinetic energy if v is the velocity of the ball at bottom then its kinetic energy at this point will be K = 12m*v2 ...(2) where m is the mass of the ball So K = W (law of the conservation of energy) 12m*v2 = m*g*L(1-cosθ ) ⇒v =2gl(1-cosθ ) ...(3) The length of the rod is 1.40 m , θ = 20° using it above equation we , get v = 2 *9.8*1.40(1-cos(20))v = 1.28m/s ⇒v =2gl(1-cosθ ) From this expression it is clear that the velocity of the ball in not depend on the mass m. so option 1 will be right.