A 263-g block is dropped onto a vertical spring with force constant k = 2.52N/cm. The block sticks to the spring, and the spring compress 11.8 cm before coming momentarily to rest. while the spring is being compressed, how much work is done:
(a) By the force of gravity
(b) By the spring?
(c) What was the speed of the block just before it hit the spring?
(d) If this initial speed of the block is doubled, what is the maximum compression of the spring? Ignore friction.
To find the answers to these questions, we can use the concepts of gravitational potential energy and spring potential energy.
(a) To find the work done by the force of gravity, we need to calculate the change in gravitational potential energy.
The gravitational potential energy can be calculated using the formula:
Potential energy = m * g * h,
where m is the mass, g is the acceleration due to gravity, and h is the height.
Given:
Mass of the block (m) = 263 g = 0.263 kg,
Acceleration due to gravity (g) = 9.8 m/s^2,
Height (h) = Compression of spring = 11.8 cm = 0.118 m.
Substituting the values into the formula, we get:
Potential energy = 0.263 kg * 9.8 m/s^2 * 0.118 m.
The work done by gravity is equal to the change in gravitational potential energy, so the work done by gravity is 0.313 J.
(b) To find the work done by the spring, we need to calculate the change in spring potential energy.
The spring potential energy can be calculated using the formula:
Potential energy = 0.5 * k * x^2,
where k is the force constant of the spring and x is the compression of the spring.
Given:
Force constant of the spring (k) = 2.52 N/cm = 2.52 * 100 N/m = 252 N/m,
Compression of the spring (x) = 11.8 cm = 0.118 m.
Substituting the values into the formula, we get:
Potential energy = 0.5 * 252 N/m * (0.118 m)^2.
The work done by the spring is equal to the change in spring potential energy, so the work done by the spring is 0.702 J.
(c) To find the speed of the block just before it hits the spring, we can use the concept of conservation of energy. The total mechanical energy of the block is equal to the sum of its kinetic energy and potential energy.
At the highest point of the block's fall just before it hits the spring, the gravitational potential energy is zero, and the total mechanical energy is equal to the kinetic energy.
The formula for kinetic energy is:
Kinetic energy = 0.5 * m * v^2, where m is the mass and v is the velocity.
Since the block is dropped, its initial speed is equal to zero. Therefore, the kinetic energy at that point is zero as well.
So, the total mechanical energy is equivalent to the gravitational potential energy at the highest point.
0.313 J = m * g * h,
0.313 J = 0.263 kg * 9.8 m/s^2 * 0.118 m.
Simplifying this equation will give us the value of h, which is the compression of the spring.
(d) If the initial speed of the block is doubled, we need to find the maximum compression of the spring. The maximum compression occurs when the block momentarily comes to rest on the spring.
To compute this, we utilize the concept of conservation of mechanical energy, where the original mechanical energy is equal to the maximum potential energy stored in the spring.
The formula for maximum potential energy stored in the spring is:
Potential energy = 0.5 * k * x_max^2, where k is the force constant of the spring and x_max is the maximum compression of the spring.
To find x_max, we can set the initial mechanical energy equal to the potential energy in the spring:
0.313 J = 0.5 * 252 N/m * x_max^2.
Simplifying this equation will give us the value of x_max, the maximum compression of the spring.
To find the amount of work done by different forces and the speed of the block just before it hits the spring, we can use the concepts of potential and kinetic energy. Let's break it down step by step:
(a) Work done by the force of gravity:
The work done by gravity can be calculated using the formula:
Work = Force * Distance * Cos(θ)
In this case, the force of gravity acting on the block is its weight, which is given by:
Weight = mass * acceleration due to gravity
Using the given mass of the block (263 g = 0.263 kg) and the acceleration due to gravity (9.8 m/s²), we can calculate the weight of the block.
Next, we need to determine the distance over which gravity does work. Since the block is dropped, it falls a distance equal to the compression of the spring (11.8 cm = 0.118 m).
The angle (θ) between the force of gravity and the direction of motion is 0° because the force of gravity acts vertically downwards, and the block falls straight down.
Therefore, we can calculate the work done by gravity using:
Work_gravity = Weight * Distance * Cos(0°)
(b) Work done by the spring:
The work done by the spring can be calculated using the formula:
Work = (1/2) * k * x²
In this case, the force constant of the spring (k) is given as 2.52 N/cm = 252 N/m. The distance over which the spring does work is the compression of the spring (11.8 cm = 0.118 m).
Using these values, we can calculate the work done by the spring using:
Work_spring = (1/2) * k * x²
(c) Speed of the block just before it hits the spring:
To determine the speed just before the block hits the spring, we can equate the potential energy of the block at the top (due to its height) to the kinetic energy just before it hits the spring.
The potential energy at the top is given by:
Potential energy = mass * gravity * height
Since the block falls from rest, the initial kinetic energy is zero.
The total mechanical energy (potential energy + kinetic energy) is conserved in the absence of any non-conservative forces like friction.
Setting the potential energy equal to the kinetic energy, we can solve for the speed just before it hits the spring using:
Potential energy = kinetic energy
mass * gravity * height = (1/2) * mass * speed²
(d) Maximum compression of the spring when the initial speed is doubled:
When the initial speed of the block is doubled, the kinetic energy at the top (before hitting the spring) becomes four times its initial value.
With this increased kinetic energy, the maximum compression of the spring can be calculated using the work-energy theorem:
Work_done = change in kinetic energy
Work_done = (1/2) * k * x_max²
change in kinetic energy = (1/2) * mass * (speed_final² - speed_initial²)
We can solve for x_max using the above equation, where the only unknown is x_max.
By following these steps, you will be able to find the answer to each question.
Ke of block = (1/2) m v^2
when spring is uncompressed
(a) work done by gravity on mass is m g h
= .263*9.81*.118 Joules
(b) work done by spring on mass is -(1/2)kx^2 = -.5(252 N/m)(.118)^2
(c) so because v = 0 at bottom
initial Ke + work done by g = work done on spring by mass
.5 (.263)v^2 + .263*9.81*.118 = .5(252 N/m)(.118)^2
solve for v
now do d :)