A block of mass m = 800 g is released from rest and slides down a frictionless track of height h = 57.0 cm. At the bottom of the track the block slides freely along a horizontal table until it hits a spring attached to a heavy, immovable wall. The spring compressed by 2.49 cm at the maximum compression. What is the value of the spring constant k?
Well, Mr. Smarty Pants, let's use our noodle here. We can start by calculating the potential energy of the block when it is at the top of the track. The formula for potential energy is mgh, where m is the mass (800 g), g is the acceleration due to gravity (which is approximately 9.8 m/s^2), and h is the height (57.0 cm).
So, the potential energy at the top of the track is:
(0.800 kg) * (9.8 m/s^2) * (0.570 m) = 4.30 J
Now, let's look at the spring. When the block hits the spring, the potential energy is transferred into the spring, causing it to compress. We can use Hooke's Law to figure out the force exerted by the spring:
F = k * x
Where F is the force, k is the spring constant (what we're trying to find), and x is the distance the spring is compressed (2.49 cm).
The potential energy in the spring can be calculated using the formula:
PE = (1/2) * k * x^2
We already know the potential energy (4.30 J) and the distance compressed (0.0249 m), so we can plug those values in:
4.30 J = (1/2) * k * (0.0249 m)^2
Now, we can solve for k:
k = 4.30 J / [(1/2) * (0.0249 m)^2]
k ≈ 13,878 N/m
And there you have it, my friend! The spring constant is approximately 13,878 N/m. Now go impress your physics teacher with your newfound knowledge!
To find the value of the spring constant k, we can use the conservation of mechanical energy. At the top of the track, the block has gravitational potential energy, and at the bottom of the track, this potential energy is fully converted into the elastic potential energy stored in the compressed spring.
Step 1: Find the gravitational potential energy of the block at the top of the track.
The formula for gravitational potential energy is given by:
Potential energy = mass x acceleration due to gravity x height
P.E. = mgh
where m is the mass, g is the acceleration due to gravity, and h is the height.
In this case, m = 800 g = 0.8 kg, g = 9.8 m/s^2, and h = 57.0 cm = 0.57 m
P.E. = 0.8 kg x 9.8 m/s^2 x 0.57 m = 4.5736 J (Joules)
Step 2: Find the elastic potential energy stored in the compressed spring.
The formula for elastic potential energy is given by:
Potential energy = (1/2) x spring constant x (spring compression)^2
P.E. = (1/2) k x x^2
where k is the spring constant and x is the maximum compression of the spring.
In this case, x = 2.49 cm = 0.0249 m
P.E. = (1/2) k x (0.0249 m)^2 = (k x 0.0249^2)/2
Step 3: Set the gravitational potential energy equal to the elastic potential energy.
4.5736 J = (k x 0.0249^2)/2
Step 4: Solve for the spring constant k.
Multiply both sides of the equation by 2:
9.1472 J = k x 0.0249^2
Divide both sides of the equation by 0.0249^2:
k = 9.1472 J / (0.0249 m)^2
Calculating this gives us:
k ≈ 1479.918 N/m
Therefore, the value of the spring constant k is approximately 1479.918 N/m.
To find the value of the spring constant (k), we can use the principles of conservation of mechanical energy.
Let's break down the problem into two parts:
1. The block sliding down the track:
When the block slides down the track, the only force acting on it is the gravitational force (mg), and since the track is frictionless, there is no work done against friction. Therefore, the initial potential energy (mgh) is converted into the kinetic energy (1/2 mv^2) at the bottom of the track.
2. The block hitting the spring:
When the block hits the spring, it compresses the spring. The potential energy of the block is converted into the elastic potential energy of the compressed spring.
Now, let's calculate the initial potential energy (mgh) and the final elastic potential energy (1/2 kx^2) where x is the compression of the spring.
Given:
Mass of the block (m) = 800 g = 0.8 kg
Height of the track (h) = 57.0 cm = 0.57 m
Compression of the spring (x) = 2.49 cm = 0.0249 m
1. Initial potential energy:
Potential Energy (PE) = mgh
PE_initial = 0.8 kg * 9.8 m/s^2 * 0.57 m
2. Final elastic potential energy:
Elastic Potential Energy (PE_spring) = 1/2 kx^2
PE_spring = 0.5 * k * (0.0249 m)^2
According to the conservation of energy principle, the initial potential energy must be equal to the final elastic potential energy. Therefore, we can set up an equation:
PE_initial = PE_spring
0.8 kg * 9.8 m/s^2 * 0.57 m = 0.5 * k * (0.0249 m)^2
Simplifying the equation and solving for k:
k = (0.8 kg * 9.8 m/s^2 * 0.57 m) / (0.5 * (0.0249 m)^2)
Calculate the value of k using the above equation, and round it to the appropriate number of significant figures.