Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

Problem 42

Chapter 10, Problem 42

A 50 g ice cube can slide up and down a frictionless 30° slope. At the bottom, a spring with spring constant 25 N/m is compressed 10 cm and used to launch the ice cube up the slope. How high does it go above its starting point?

Verified step by step guidance

Convert all given quantities to SI units. The mass of the ice cube is 50 g = 0.05 kg, the spring constant is 25 N/m, and the spring compression is 10 cm = 0.1 m.

Calculate the elastic potential energy stored in the spring using the formula:

U_{spring} = \frac{1}{2} k x^{2}

, where

k

is the spring constant and

x

is the compression distance.

Apply the principle of conservation of energy. The elastic potential energy of the spring is converted into gravitational potential energy as the ice cube moves up the slope. The gravitational potential energy is given by:

U_{gravity} = mgh

, where

m

is the mass,

g

is the acceleration due to gravity (9.8 m/s²), and

h

is the height gained.

Relate the height

h

to the distance traveled along the slope using trigonometry. Since the slope angle is 30°, the height is given by:

h = d sin(30°)

, where

d

is the distance traveled along the slope.

Set the elastic potential energy equal to the gravitational potential energy and solve for

h

. The equation becomes:

\frac{1}{2} k x^{2} = mgh

. Substitute the known values for

k

x

m

, and

g

to find

h

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the potential energy stored in the compressed spring is converted into gravitational potential energy as the ice cube moves up the slope. Understanding this concept is crucial for calculating the maximum height the ice cube reaches.

Spring Potential Energy

Spring potential energy is the energy stored in a compressed or stretched spring, calculated using the formula PE_spring = 1/2 k x², where k is the spring constant and x is the compression or extension from its equilibrium position. In this problem, the spring constant is 25 N/m and the compression is 0.1 m, which allows us to determine the initial energy available to launch the ice cube.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, given by the formula PE_gravity = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above a reference point. This concept is essential for calculating how high the ice cube rises after being launched by the spring, as it will convert the spring's potential energy into gravitational potential energy.

Recommended video:

Guided course

06:35

Gravitational Potential Energy

Related Practice

Textbook Question

You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 10-m-high hill, then descends 15 m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 400 kg. For safety reasons, the spring constant should be 10% larger than the minimum needed for the car to just make it over the top. What is the maximum speed of a 350 kg car if the spring is compressed the full amount?

views

Textbook Question

A horizontal spring with spring constant 100 N/m is compressed 20 cm and used to launch a 2.5 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Use work and energy to find how far the box slides across the rough surface before stopping.

views

Textbook Question

A block of mass m slides down a frictionless track, then around the inside of a circular loop-the-loop of radius R . From what minimum height h must the block start to make it around without falling off? Give your answer as a multiple of R.

Textbook Question

A 50 g mass is attached to a light, rigid, 75-cm-long rod. The other end of the rod is pivoted so that the mass can rotate in a vertical circle. What speed does the mass need at the bottom of the circle to barely make it over the top of the circle?

views

Textbook Question

How much work is done by the environment in the process shown in FIGURE EX10.39? Is energy transferred from the environment to the system or from the system to the environment?

Textbook Question

A cable with 20.0 N of tension pulls straight up on a 1.50 kg block that is initially at rest. What is the block's speed after being lifted 2.00 m? Solve this problem using work and energy.