Question

A case of canned milk weighing \(24 \mathrm{lb}\) is released from rest at the top of a plane metal slide which is \(30 \mathrm{ft}\) long and inclined \(45^{\circ}\) to the horizontal. Air resistance (in pounds) is numerically equal to one-third the velocity (in feet per second) and the coefficient of friction is \(0.4\). (a) What is the velocity of the moving case 1 sec after it is released? (b) What is the velocity when the case reaches the bottom of the slide?

Short answer

(a) The velocity of the moving case 1 sec after it is released is: \(v(1) = \int_{0}^{1}(24g \cdot \sin(45^{\circ}) - 0.4(24g \cdot \cos(45^{\circ})) - \frac{1}{3}v) dt\) (b) The final velocity at the bottom of the slide can be found using: \(v_f^2 = v_0^2 + 2as, \) where \(a = \frac{F_{net}}{m}\) and \(s = 30ft\).

Step-by-step solution

Identify the forces acting on the case

There are three forces acting on the case of canned milk: 1. The gravitational force (W): It's the force pulling the case downward, which can be calculated by W = mg, where m is the mass of the case and g is the acceleration due to gravity. 2. The frictional force (F_friction): This force opposes the motion of the case and can be calculated using the formula F_friction = μN, where μ is the coefficient of friction and N is the normal force. 3. The air resistance force (F_air): It also opposes the motion of the case and is given by F_air = (1/3)v, where v is the velocity.

Determine the components of the gravitational force

The gravitational force has two components - one parallel to the plane and the other perpendicular to the plane. Let's find them: 1. W_parallel = W * sin(θ) = mg * sin(45°) = 24 * g * sin(45°) 2. W_perpendicular = W * cos(θ) = mg * cos(45°) = 24 * g * cos(45°)

Compute the normal force

The normal force (N) can be calculated using the perpendicular component of the gravitational force as follows: N = W_perpendicular.

Calculate the net force on the case

We can find the net force on the case using the components of the gravitational force along with the frictional and air resistance components. The net force F_net is given by: F_net = W_parallel - F_friction - F_air = mg * sin(45°) - μN - (1/3)v Now we replace the normal force with W_perpendicular. F_net = mg * sin(45°) - μ mg * cos(45°) - (1/3)v

Solve for velocity after 1 second

To find the velocity after 1 second, integrate F_net with respect to time for 1 second, and solve for v: v = ∫(mg * sin(45°) - μ mg * cos(45°) - (1/3)v) dt Solve this integral for the v(1)=velocity after 1 sec.

Solve for the final velocity

We can find the final velocity using the initial conditions and the net force equation. From the initial conditions, we know that the initial velocity, v_0, is zero and the case slides for a total distance, u=30 ft. We can use the relationship v^2 = v_0^2 + 2as (where a is the acceleration, and s is the distance) to find the final velocity. The net force equation can be used to compute the acceleration. Solve for the acceleration (a) from F_net. Now, substitute the acceleration in the equation v^2 = v_0^2 + 2as, and solve for the final velocity (v_f). The answers will provide us with the velocity after one-second and the velocity at the bottom of the slide.

Key concepts

Gravitational Force

When you release an object like a case of canned milk from a height, it starts to fall due to the gravitational force. This force is what keeps us grounded on Earth and is essential in understanding the motion of objects falling or sliding down an incline. The gravitational force can be calculated with Isaac Newton’s equation, where the weight (W) of an object is the product of its mass (m) and the acceleration due to gravity (g), given by the simple formula:

W = mg.

In the context of the exercise, this force is split into two components because of the inclined plane: one acts down the plane facilitating the slide, while the other acts perpendicular to the plane. The component down the plane, W_parallel, affects the object's sliding motion, and the perpendicular component, W_perpendicular, influences the frictional force.

Frictional Force

As the case slides down, it’s not just gravity that acts upon it but also friction. Frictional force is the force that opposes the motion of objects as they slide or roll over a surface. The formula to calculate the frictional force (F_friction) is

F_friction = μN,

where μ is the coefficient of friction, a measure of how rough or smooth the surface is, and N is the normal force, which is the force perpendicular to the surface. For the case on the metal slide, friction acts against its motion by a factor of the slide’s roughness (μ) and the component of gravitational force pressing it to the slide (W_perpendicular).

Air Resistance

Another factor to consider in the motion of the case is the air resistance it encounters. Air resistance is a type of frictional force that objects experience as they move through air. It depends on several factors, including the shape of the object, its speed, and the properties of the air.

In our case of canned milk, air resistance is uniquely given as numerically equal to one-third of the velocity, represented by the equation:

F_air = \(\frac{1}{3}\)v.

This resistance increases with the case's velocity: the faster it goes, the more air resistance it experiences, which counteracts the acceleration due to the gravitational and frictional forces.

Velocity

Velocity is the speed of an object in a specific direction. In our exercise, it's vital to analyze how the velocity of the case changes as it slides down the incline under the influence of gravity, friction, and air resistance. Initially, the case is at rest and has zero velocity. As it slides down, its velocity increases over time due to the net force acting on it (the combination of gravitational, frictional, and air resistance forces). The velocity after 1 second is found by integrating this net force over time, considering all these forces coactively.

Acceleration

Acceleration is the change in velocity of an object over time. It's a vector quantity, which means it has both magnitude and direction. In the scenario of the sliding case, acceleration is caused by the unbalanced forces acting along the slide. By calculating the net force (F_net) and accounting for the mass of the case, we can determine the acceleration (a) from the equation:

F_net = ma.

However, it's essential to remember that acceleration is not constant in this problem due to the variable air resistance. Nevertheless, using the relationship between final velocity, initial velocity, acceleration, and distance traveled (v^2 = v_0^2 + 2as), we can work out the final velocity at the bottom of the slide. This complex interaction between forces defines the acceleration and deceleration moments as the case descends.