Problem 79 - University Physics with Modern Physics

Question

You are working for a shipping company. Your job is to stand at the bottom of a \(8.0-\mathrm{m}\) -long ramp ramp that is inclined at \(37^{\circ}\) above the horizontal. You grab packages off a conveyor belt and propel them up the ramp. The coefficient of kinetic friction between the packages and the ramp is \(\mu_{\mathrm{k}}=0.30\) . (a) What speed do you need to give a package at the bottom of the ramp so that it has zero speed at the top of the ramp? (b) Your coworker is supposed to grab the packages as they arrive at the top of the ramp, but she misses one and it slides back down. What is its speed when it returns to you?

Step-by-Step Solution

Answer

(a) The initial speed required is 6.25 m/s. (b) The returning speed is 9 m/s.

Step 1: Understand the Problem

We need to find the speed to give to packages so they reach rest at the top, considering friction, and find the returning speed if a package slides back due to gravity.

Step 2: Sketch the Ramp Situation

Visualize and sketch the ramp, noting the inclines and lengths. Mark the forces acting on the package, such as gravitational force, normal force, and frictional force.

Step 3: Find Total Work Done on the Package

Identify that the package moves 8 meters up the ramp against gravity and friction. The work done by you should equal the sum of the work done against gravity and work done against friction.

Step 4: Set the Energy Equation for Part (a)

Use energy conservation and work-energy principle:Initial kinetic energy = Work done against gravity + Work done against friction.Using formulas:- Gravitational force component along the ramp: \(mg\sin\theta\)- Frictional force: \(\mu_k mg\cos\theta\)And equations for work:- Work done against gravity \(= mgh\)- Work done against friction \(= \mu_k mg d \cos\theta\).

Step 5: Calculate the Required Initial Speed

Combine the energy equations: \( \frac{1}{2}mv^2 = mgh + \mu_k mg d \cos\theta \)Plug in values \(h = 8\sin(37^{\circ})\) and solve for \(v\) that reaches zero speed at the top.

Step 6: Determine Work Done Sliding Back (Part b)

For the slide down, use energy conservation: Potential energy at the top converts to kinetic energy at the bottom less work done against friction.

Step 7: Calculate Speed When Returned

Use conversion from potential to kinetic energy:\(mgh = \frac{1}{2}mv^2 + \mu_k mg d \cos\theta\) Solve for the sliding speed \(v\) when it reaches the bottom.

Key Concepts

Kinematic Friction

Kinematic friction plays a significant role in inclined plane motion. Whenever an object slides down a surface, friction opposes the motion and can slow it down. This force is defined by the equation:

• Frictional force = \( \mu_k \cdot N \)

where \( \mu_k \) is the coefficient of kinetic friction, and \( N \) is the normal force.
On an inclined plane, the normal force is less than the object's weight because it's perpendicular to the slope. If the angle of inclination is \( \theta \), the normal force is calculated by \( N = mg \cos \theta \). This calculation shows that the frictional force is directly proportional to the cosine of the angle.In our exercise, the coefficient of kinetic friction is 0.30. This means that a significant fraction of the gravitational pull is countered by friction. The work you do to move a package up is affected by this friction, which needs to be overcome to keep the package moving.

Work-Energy Principle

The work-energy principle is critical to understanding how energy is transferred when you push the package up the ramp. This principle states that the work done on an object results in a change in its kinetic energy.

• Work done = Change in kinetic energy

In mathematical terms, this is expressed as: \[\frac{1}{2} mv^2 = mgh + \mu_k mgd \cos \theta\]Here,

• \( \frac{1}{2} mv^2 \) is the initial kinetic energy of the package.
• \( mgh \) is the work done against gravity, where \( h \) is the vertical height of the ramp.
• \( \mu_k mgd \cos \theta \) is the work done against friction as the package moves up the ramp.

By using this equation, you can determine how much initial velocity is necessary for the package to reach the top with zero velocity, overcoming both gravity and friction along the way.

Conservation of Energy

The conservation of energy is another vital concept when analyzing the motion of objects on an incline. This principle states that energy cannot be created or destroyed, only transformed from one form to another.When the package makes its way back down the ramp, its potential energy at the top is converted into kinetic energy at the bottom, minus the energy lost due to friction. The relevant equation is:\[mgh = \frac{1}{2} mv^2 + \mu_k mgd \cos \theta\]In this expression,

• \( mgh \) represents the potential energy at the top of the ramp.
• \( \frac{1}{2} mv^2 \) refers to the kinetic energy it has when reaching the bottom.
• \( \mu_k mgd \cos \theta \) accounts for the work done against friction.

With this relationship, you can predict the speed of a package when it returns to the bottom of the ramp, understanding how friction and height influence its energy transformation.