A student places her 500 g physics book on a frictionless table. She pushes the book
against a spring, compressing the spring by 4.0 cm, then releases the book. What is the
book’s speed as it slides away? The spring constant is 1250 N/m.
1
Well, it sounds like this book has quite the spring in its step! Okay, let's get down to business. To find the speed of the book as it slides away, we'll need to use a little bit of physics (but don't worry, I'll keep it fun!).
First, let's determine the potential energy stored in the spring when it's compressed. The formula for that is:
Potential energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring from its equilibrium position. In this case, the displacement is 4.0 cm, or 0.04 m, and the spring constant is 1250 N/m. So we have:
Potential energy = (1/2) * 1250 N/m * (0.04 m)^2
Now let's remember that energy is always conserved. When the book is released, all of the potential energy stored in the spring is converted into kinetic energy of the book. So:
Potential energy = Kinetic energy
The formula for kinetic energy is:
Kinetic energy = (1/2) * m * v^2
where m is the mass of the book and v is its velocity. We can rearrange this equation to solve for v:
v = sqrt((2 * potential energy) / m)
Since the book has a mass of 500 g, that's 0.5 kg. Plugging in the values for potential energy and mass, we get:
v = sqrt((2 * (0.5 N * 0.04 m)^2) / 0.5 kg)
Now, let's crunch the numbers and calculate the book's speed!
v = sqrt((2 * 0.5 N * 0.04 m)^2) / 0.5 kg)
v = sqrt((2 * 0.08 J) / 0.5 kg)
v = sqrt(0.16 J / 0.5 kg)
v = sqrt(0.32 m^2/s^2)
v ≈ 0.57 m/s
So there you have it! The book's speed as it slides away is approximately 0.57 m/s. I hope that answer brings a little bounce to your step!
To calculate the book's speed as it slides away, we can use the principle of conservation of energy.
Step 1: Determine the potential energy stored in the compressed spring.
The potential energy stored in a spring is given by the equation:
Potential Energy = 0.5 * spring constant * (compression distance)^2
Given:
Spring constant = 1250 N/m
Compression distance = 0.04 m (4.0 cm)
Potential Energy = 0.5 * 1250 N/m * (0.04 m)^2
Potential Energy = 0.5 * 1250 N/m * 0.0016 m^2
Potential Energy = 1 J (Joule)
Step 2: Determine the kinetic energy of the book as it slides away.
According to the principle of conservation of energy, the potential energy stored in the spring will be converted into kinetic energy as the book slides away.
Since the potential energy is equal to the kinetic energy, we have:
Kinetic Energy = Potential Energy
Kinetic Energy = 1 J
Step 3: Determine the velocity of the book.
The kinetic energy of an object can be calculated using the equation:
Kinetic Energy = 0.5 * mass * velocity^2
Given:
Mass of the book = 500 g = 0.5 kg
0.5 * velocity^2 = 1 J
velocity^2 = 2 J/kg
velocity = sqrt(2 J/kg)
Using a calculator, we find the velocity to be approximately 1.41 m/s.
Therefore, the book's speed as it slides away is approximately 1.41 m/s.
To find the book's speed as it slides away, we can use the principle of conservation of mechanical energy. The mechanical energy of the system (book + spring) remains constant throughout the process.
First, let's find the potential energy stored in the spring when it is compressed by 4.0 cm. The potential energy stored in a spring can be calculated using the formula:
Potential energy (PE) = (1/2) * k * x^2
Where k is the spring constant and x is the displacement (4.0 cm = 0.04 m in this case).
PE = (1/2) * 1250 N/m * (0.04 m)^2
= 0.5 * 1250 N/m * 0.0016 m^2
= 1 N * 0.0016 m^2
= 0.0016 Nm
= 0.0016 J
Now, let's calculate the kinetic energy (KE) of the book as it slides away. Since the book starts at rest (with no initial velocity), all the potential energy stored in the spring is converted into kinetic energy.
KE = PE
= 0.0016 J
The kinetic energy can be expressed using the formula:
KE = (1/2) * m * v^2
Where m is the mass of the book (500 g = 0.5 kg in this case) and v is its velocity.
Rearranging the formula, we get:
v^2 = (2 * KE) / m
Substituting the values:
v^2 = (2 * 0.0016 J) / 0.5 kg
= 0.0032 J / 0.5 kg
= 0.0064 J/kg
Taking the square root of both sides, we find:
v = sqrt(0.0064 J/kg)
≈ 0.08 m/s
Therefore, the book's speed as it slides away is approximately 0.08 m/s.
book KineticEnergy=Max PE of spring
1/2 m v^2=1/2 k x^2
x= .04m, m=1/2 kg
solve for velocity v