A student places her 500 g physics book on a frictionless table. She pushes the book against a spring, compressing the spring by 4.40 cm, then releases the book. What is the book's speed as it slides away? The spring constant is 1950 N/m.
1/2 mv^2 = 1/2 kx^2
To find the book's speed as it slides away, we can use the principles of conservation of mechanical energy. Initially, the book possesses potential energy stored in the compressed spring, which is then converted to kinetic energy as the book slides away.
1. Calculate the potential energy stored in the compressed spring:
The potential energy stored in a spring is given by the formula:
Potential energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring (compression in this case).
In this problem, k = 1950 N/m and x = 4.40 cm = 0.044 m
Potential energy = (1/2) * 1950 N/m * (0.044 m)^2
2. Equate the potential energy to the kinetic energy of the book as it slides away:
According to the conservation of mechanical energy,
Potential energy = Kinetic energy
(1/2) * 1950 N/m * (0.044 m)^2 = (1/2) * mass * speed^2
In this problem, the mass of the book = 500 g = 0.5 kg
Plugging in the values,
(1/2) * 1950 N/m * (0.044 m)^2 = (1/2) * 0.5 kg * speed^2
3. Solve for the speed of the book:
Divide both sides of the equation by (1/2) * 0.5 kg to isolate the speed.
speed^2 = [(1/2) * 1950 N/m * (0.044 m)^2] / [(1/2) * 0.5 kg]
Simplifying,
speed^2 = (1950 N/m * (0.044 m)^2) / 0.5 kg
Take the square root of both sides to find the speed.
speed = √[(1950 N/m * (0.044 m)^2) / 0.5 kg]
Now, calculate the speed using a calculator or by using the formula speed = √(a/b) where a = (1950 N/m * (0.044 m)^2) and b = 0.5 kg.
Hence, the book's speed as it slides away can be calculated using the given formula.
To find the book's speed as it slides away, we can use the principle of conservation of mechanical energy. At the moment the book is released, the potential energy stored in the compressed spring is converted into the kinetic energy of the book as it slides away.
First, let's calculate the potential energy stored in the spring when it's compressed by 4.40 cm. The potential energy stored in a spring can be calculated using the formula:
Potential Energy = (1/2) * k * Δx^2
where k is the spring constant and Δx is the displacement (change in position).
In this case, the spring constant is given as 1950 N/m and the displacement is 4.40 cm. Let's convert the displacement to meters:
Δx = 4.40 cm = 0.0440 m
Now we can calculate the potential energy:
Potential Energy = (1/2) * (1950 N/m) * (0.0440 m)^2
Next, we equate the potential energy of the spring to the kinetic energy of the book as it slides away:
Potential Energy = Kinetic Energy
(1/2) * (1950 N/m) * (0.0440 m)^2 = (1/2) * m * v^2
where m is the mass of the book and v is its velocity (speed).
In this case, the mass of the book is given as 500 g. Let's convert the mass to kilograms:
m = 500 g = 0.500 kg
Now we can solve for v:
(1/2) * (1950 N/m) * (0.0440 m)^2 = (1/2) * (0.500 kg) * v^2
Simplifying the equation:
v^2 = ( (1/2) * (1950 N/m) * (0.0440 m)^2 ) / (0.500 kg)
v^2 = 9.9036 m^2/s^2
Taking the square root of both sides:
v = √(9.9036 m^2/s^2)
v ≈ 3.15 m/s
Therefore, the book's speed as it slides away is approximately 3.15 m/s.