A crane of mass \(M=1.00 \cdot 10^{4} \mathrm{~kg}\) lifts a wrecking ball of mass \(m=1200 .\) kg directly upward. a) Find the magnitude of the normal force exerted on the crane by the ground while the wrecking ball is moving upward at a constant speed of \(v=1.00 \mathrm{~m} / \mathrm{s}\). b) Find the magnitude of the normal force if the wrecking ball's upward motion slows at a constant rate from its initial speed \(v=1.00 \mathrm{~m} / \mathrm{s}\) to a stop over a distance \(D=0.250 \mathrm{~m}\)

Understanding Newton's second law is crucial when exploring the dynamics of motion. This law, often written as \( F_{net} = ma \), relates the net force (\( F_{net} \)) acting on an object to its mass (\( m \)) and acceleration (\( a \)). It's the backbone of classical mechanics and helps students analyze the forces in action, whether it's a car accelerating down a highway or a wrecking ball hoisted by a crane.

For instance, in our exercise, when the wrecking ball is lifted at a constant speed, its acceleration is zero. This leads to the conclusion that the net force is zero as well, meaning the forces acting on the system are in equilibrium. Therefore, the normal force exerted by the ground equals the gravitational force acting on the crane and the wrecking ball together. When the lifting speed changes, we then need to calculate the new acceleration to find the updated net force and, consequently, the new normal force the ground must exert.

A constant velocity indicates that an object is moving at a steady speed in a single direction. Importantly, 'constant velocity' also implies that there is no acceleration—since acceleration is the rate of change of velocity. When an object has a constant velocity, as cited in our wrecking ball example moving upward at \(1.00\,\mathrm{m/s}\), the forces acting upon it are balanced. This equilibrium means that the upward force (the tension in the crane's cable) directly opposes the downward force of gravity, resulting in a net force of zero.

Why Does Constant Velocity Equate to Zero Net Force?

In this state, Newton’s first law (often termed inertia) comes into play. It states that an object in motion at a constant velocity will stay in motion, unless acted upon by an external force. For our crane and wrecking ball, the absence of acceleration assures us that the net forces are indeed balanced, affording us the simplicity of calculating the normal force without the complication of unbalanced forces.

The acceleration due to gravity, denoted by \(g\), is the acceleration that the Earth imparts to objects on or near its surface. Standard gravity is generally accepted to be \(9.81\,\mathrm{m/s^2}\) on Earth's surface. This constant acceleration affects any object under free fall and also contributes to the ‘weight’ (gravitational force) experienced by objects.

How Gravity Affects Our Calculations

Gravity plays a leading role in determining the normal force in our problem. Even when the crane holds the wrecking ball at a steady height or moves it at constant velocity, gravity imposes a continuous downward force equal to the mass of the crane and wrecking ball times \(g\). The normal force has to counteract this to maintain equilibrium, as it's the force exerted by a surface to support the weight of an object resting on it.

Kinematic equations let us determine the motion characteristics of objects when acceleration is constant. The basic set includes formulas that relate displacement, initial and final velocities, acceleration, and time. These equations are powerful tools for analyzing motion, and they are particularly handy when figuring out problems involving objects accelerating under the influence of gravity.

In the situation where the wrecking ball comes to a stop over a certain distance, we use a kinematic equation to connect its initial velocity, the stopping distance, and the unknown acceleration. Knowing the final velocity is zero allows us to rearrange the equation to solve for the acceleration, which then feeds into Newton’s second law to determine the net force during this deceleration. This net force, when coupled with gravity’s pull, gives us insight into the change in the normal force during this phase of motion.