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Chapter 6: Problem 26

A block of ice with mass 2.00 \(\mathrm{kg}\) slides 0.750 \(\mathrm{m}\) down an inclined plane that slopes downward at an angle of \(36.9^{\circ}\) below the horizontal. If the block of ice starts from rest, what is its final speed? You can ignore friction.

Short Answer

Expert verified
The final speed of the block is approximately 2.97 m/s.

Step by step solution

01

Identify the Forces and Initial Conditions

We are given a block of ice with a mass of 2.00 kg sliding down an inclined plane. The block starts from rest and slides a distance of 0.750 m down the plane at an angle of 36.9掳 to the horizontal. Friction is ignored.
02

Resolve the Force of Gravity

The force of gravity acting on the block can be resolved into components parallel and perpendicular to the incline.The component of gravitational force parallel to the incline: \[ F_{\parallel} = m \cdot g \cdot \sin(\theta) \]where \( m = 2.00 \ \mathrm{kg} \), \( g = 9.81 \ \mathrm{m/s^2} \), and \( \theta = 36.9^\circ \).
03

Calculate the Parallel Force

Substitute in the values to calculate the parallel component of the force:\[ F_{\parallel} = 2.00 \times 9.81 \times \sin(36.9^\circ) \]\[ F_{\parallel} \approx 2.00 \times 9.81 \times 0.600 \approx 11.772 \ \mathrm{N} \]
04

Calculate the Work Done by Gravity

The work done by the force of gravity is equal to this force component times the distance:\[ W = F_{\parallel} \times s \]\[ W = 11.772 \times 0.750 \approx 8.829 \ \mathrm{J} \]
05

Use the Work-Energy Principle

According to the work-energy principle, the work done by the net force is equal to the change in kinetic energy:\[ W = \Delta KE = \frac{1}{2} mv_f^2 - \frac{1}{2} mv_i^2 \]Since the block starts from rest, \( v_i = 0 \) and thus:\[ 8.829 = \frac{1}{2} \times 2.00 \times v_f^2 \]
06

Solve for the Final Speed

Rearrange the equation to solve for \( v_f^2 \):\[ v_f^2 = \frac{2 \times 8.829}{2.00} \approx 8.829 \]Taking the square root gives the final speed:\[ v_f = \sqrt{8.829} \approx 2.97 \ \mathrm{m/s} \]

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Key Concepts

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

Inclined Plane
An inclined plane is simply a flat surface set at an angle, other than 90 degrees, against a horizontal surface. Inclined planes make it easier to move objects upwards by spreading the needed effort over a longer distance. For our exercise, the inclined plane is sloped at an angle of 36.9掳. This means the plane is not extremely steep, which affects how gravity acts on the block.
Although inclined planes reduce the force necessary to raise an object, the object will still experience gravity pulling it parallel to the slope.
  • The angle of the incline (36.9掳 in this case) will determine the direction and magnitude of gravitational force components.
  • Inclined planes are ideal for illustrating fundamental physics concepts, including forces and motion.
Work-Energy Principle
The work-energy principle is a core concept in physics stating that the work done on an object is equal to the change in its kinetic energy. It simplifies problems by linking kinetic energy directly with forces exerted over a distance.In the given problem, the work done on the ice block as it slides down the plane helps to calculate its change in kinetic energy.
  • Initial kinetic energy of the block is zero, as it starts from rest.
  • The work done by gravity is used to increase the block's kinetic energy.
This principle allowed us to determine the final speed of the ice block using the work equation:\[W = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\]
Gravitational Force
Gravitational force is the force by which a planet or other such body draws objects toward its center. It is a key factor in determining the motion of an object on an inclined plane.
  • The gravitational force can be split into perpendicular and parallel components relative to the inclined plane.
  • The perpendicular component pulls the object toward the plane, while the parallel component pulls it down the slope.
For our block of ice:\[F_{\parallel} = mg \sin(\theta)\]This parallel component is pivotal for calculating the work done by gravity as the block slides down the inclined plane.
Kinetic Energy
Kinetic energy is the energy of motion. An object inevitably gains kinetic energy when work is done on it, accelerating it or changing its state of motion.At the start, our block of ice has zero kinetic energy since it is at rest. As it slides down the inclined plane, the gravitational force does work on it. This work converts potential energy into kinetic energy, giving speed to the block.
  • The formula for kinetic energy is: \( KE = \frac{1}{2}mv^2 \)
  • The change in kinetic energy of the block provides an easy way to determine its final speed after sliding a certain distance.
Remember, kinetic energy depends on two factors: mass and velocity.
Newton's Laws
Newton's Laws of Motion form the foundation for understanding forces and motions.
  • First Law: An object remains at rest or in uniform motion unless acted upon by a force. Here, the block starts moving due to the gravitational force acting parallel to the incline.
  • Second Law: Describes how the velocity of an object changes when it is subjected to an external force. In our example, the component of gravity along the incline affects the acceleration of the block.
  • Third Law: For every action, there is an equal and opposite reaction. While not directly addressed in this exercise, it's essential in many other aspects of inclined plane motion.
These laws explain why the block accelerates down the slope and why we can use the work-energy principle to find its final speed.

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Most popular questions from this chapter

An object is attracted toward the origin with a force given by \(F_{x}=-k / x^{2} .\) (Gravitational and electrical forces have this distance dependence.) (a) Calculate the work done by the force \(F_{x}\) when the object moves in the \(x\) -direction from \(x_{1}\) to \(x_{2}\) . If \(x_{2}>x_{1}\) , is the work done by \(F_{x}\) positive or negative? (b) The only other force acting on the object is a force that you exert with your hand to move the object slowly from \(x_{1}\) to \(x_{2}\) . How much work do you do? If \(x_{2}>x_{1},\) is the work you do positive or negative? (c) Explain the similarities and differences between your answers to parts (a) and (b).

To stretch a spring 3.00 \(\mathrm{cm}\) from its unstretched length, 12.0 \(\mathrm{J}\) of work must be done. (a) What is the force constant of this spring? (b) What magnitude force is needed to stretch the spring 3.00 \(\mathrm{cm}\) from its unstretched length? (c) How much work must be done to compress this spring 4.00 \(\mathrm{cm}\) from its unstretched length, and what force is needed to streteh it this distance?

Automotive Power II. (a) If 8.00 hp are required to drive a \(1800-\mathrm{kg}\) automobile at 60.0 \(\mathrm{km} / \mathrm{h}\) on a level road, what is the total retarding force due to friction, air resistance, and so on? (b) What power is necessary to drive the car at 60.0 \(\mathrm{km} / \mathrm{h}\) up a 10.0\(\%\) grade (a hill rising 10.0 \(\mathrm{m}\) vertically in 100.0 \(\mathrm{m}\) horizontally)? (c) What power is necessary to drive the car at 60.0 \(\mathrm{km} / \mathrm{h}\) down a 1.00\(\%\) grade? (d) Down what percent grade would the car coast at 60.0 \(\mathrm{km} / \mathrm{h} ?\)

On an essentially frictionless, horizontal ice rink, a skater moving at 3.0 \(\mathrm{m} / \mathrm{s}\) encounters a rough patch that reduces her speed by 45\(\%\) due to a friction force that is 25\(\%\) of her weight. Use the work- energy theorem to find the length of this rough patch.

The total consumption of electrical energy in the United States is about 1.0 \(\times 10^{19} \mathrm{J}\) per year. (a) What is the average rate of electrical energy consumption in watts? (b) The population of the United States is about 300 million people. What is the average rate of electrical energy consumption per person? (c) The sun transfers energy to the earth by radiation at a rate of approximately 1.0 \(\mathrm{kW}\) per square meter of surface. If this energy could be collected and converted to electrical energy with 40\(\%\) efficiency, how great an area (in square kilometers) would be required to collect the electrical energy used in the United States?
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