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Chapter 6: Problem 23

You are a member of an Alpine Rescue Team. You must project a box of supplies up an incline of constant slope angle \(\alpha\) so that it reaches a stranded skier who is a vertical distance \(h\) above the bottom of the incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient \(\mu_k\). Use the work\(-\) energy theorem to calculate the minimum speed you must give the box at the bottom of the incline so that it will reach the skier. Express your answer in terms of \(g\), \(h\), \(\mu_k\), and \(\alpha\).

Short Answer

Expert verified
The minimum speed needed is \( v_i = \sqrt{2gh (1 + \mu_k \cot(\alpha))} \).

Step by step solution

01

Apply Work-Energy Theorem

According to the work-energy theorem, the work done by all forces is equal to the change in kinetic energy. For the box, the change in kinetic energy (\( \Delta KE\) ) is given by \( \frac{1}{2} mv^2_f - \frac{1}{2} mv^2_i \). Since the box comes to rest at the top, \(v_f = 0\). Hence, \(\Delta KE = - \frac{1}{2} mv^2_i \).
02

Calculate Work Done Against Gravity

As the box moves up the incline, it gains gravitational potential energy equivalent to \( mgh \). This energy must be provided by the initial kinetic energy.
03

Calculate Work Done Against Friction

Friction does negative work on the box as it moves up the incline. The work done by friction \(W_f\) is given by \( \mu_k mg \cos(\alpha) d \), where \( d \) is the distance traveled along the incline. Use geometry to express \( d = \frac{h}{\sin(\alpha)} \), thus \(W_f = \mu_k mg \frac{h \cos(\alpha)}{\sin(\alpha)}\).
04

Equate Work-Energy Balance

Set up the equation for the work-energy balance: \[0 = - \frac{1}{2} mv_i^2 + mgh + \mu_k mg \frac{h \cos(\alpha)}{\sin(\alpha)}\].
05

Solve for Initial Velocity

Rearrange the equation to solve for the initial velocity \( v_i \): \[ \frac{1}{2} mv_i^2 = mgh (1 + \mu_k \cot(\alpha)) \].Solve for \( v_i \): \[ v_i = \sqrt{2gh (1 + \mu_k \cot(\alpha))} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Potential Energy
Gravitational potential energy is a form of energy associated with the height of an object. When you lift an object against the force of gravity, you're doing work on it, increasing its potential to do work when it falls back down. For our exercise, as the box of supplies is projected up an incline, it moves to a greater height, thereby increasing its gravitational potential energy.
This energy is calculated using the formula:
  • \( mgh \)
  • where \( m \) is the mass of the box, \( g \) is the acceleration due to gravity, and \( h \) is the vertical distance covered.
In this scenario, the initial kinetic energy imparted must be enough to allow the box to reach the desired height by overcoming both gravity and friction.
Kinetic Friction
Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. Unlike static friction, which prevents an object from moving, kinetic friction acts on objects that are already in motion. The force of kinetic friction can be calculated using the formula:
  • \( f_k = \mu_k N \)
  • where \( \mu_k \) is the coefficient of kinetic friction and \( N \) is the normal force.
In our inclined plane problem, kinetic friction works against the box as it slides up the slope, doing negative work by taking away some of the box's energy. This force depends on the box's weight and the angle of the incline, which affects the normal force experienced by the box.
Inclined Plane
An inclined plane is a flat surface tilted at an angle, and it's a classic physics scenario used to analyze forces and energy. In this exercise, the incline not only affects how gravity pulls on the box but also the normal force impacting kinetic friction. The angle \( \alpha \) affects how much of the gravitational force acts parallel to the plane.
The characteristics of the inclined plane are crucial as they determine how much work is done against gravity and friction. The geometry of the plane helps in:
  • Computing the distance \( d \) using \( d = \frac{h}{\sin(\alpha)} \), which is key to calculating the work done.
  • Defining the components of forces acting on the box.
Initial Velocity
Initial velocity is the speed given to an object to start its motion. For this problem, determining the required initial velocity is essential to ensure the box reaches the skier. The initial speed must overcome the energy losses due to gravitational potential energy gain and kinetic friction.
Using the work-energy theorem, we find that the minimum initial velocity \( v_i \) needed is: \[ v_i = \sqrt{2gh (1 + \mu_k \cot(\alpha))} \]This equation results from balancing the energies involved:
  • The loss in kinetic energy as the box climbs.
  • The increase in potential energy.
  • The work done against friction.
The initial velocity ensures that the starting kinetic energy is sufficient for the box to overcome all these energy demands and safely reach its target.

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Most popular questions from this chapter

(a) In the Bohr model of the atom, the ground state electron in hydrogen has an orbital speed of 2190 km/s. What is its kinetic energy? (Consult Appendix F.) (b) If you drop a 1.0-kg weight (about 2 lb) from a height of 1.0 m, how many joules of kinetic energy will it have when it reaches the ground? (c) Is it reasonable that a 30-kg child could run fast enough to have 100 J of kinetic energy?

At a waterpark, sleds with riders are sent along a slippery, horizontal surface by the release of a large compressed spring. The spring, with force constant \(k\) = 40.0 N/cm and negligible mass, rests on the frictionless horizontal surface. One end is in contact with a stationary wall. A sled and rider with total mass 70.0 kg are pushed against the other end, compressing the spring 0.375 m. The sled is then released with zero initial velocity. What is the sled's speed when the spring (a) returns to its uncompressed length and (b) is still compressed 0.200 m?

The Grand Coulee Dam is 1270 m long and 170 m high. The electrical power output from generators at its base is approximately 2000 MW. How many cubic meters of water must flow from the top of the dam per second to produce this amount of power if 92% of the work done on the water by gravity is converted to electrical energy? (Each cubic meter of water has a mass of 1000 kg.)

While hovering, a typical flying insect applies an average force equal to twice its weight during each downward stroke. Take the mass of the insect to be 10 g, and assume the wings move an average downward distance of 1.0 cm during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect.

A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a 16.0-cm strip of the donated aorta reveal that it stretches 3.75 cm when a 1.50-N pull is exerted on it. (a) What is the force constant of this strip of aortal material? (b) If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is 1.14 cm, what is the greatest force it will be able to exert there?
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