A 5.80 kg object on a horizontal frictionless surface is attached to a spring with spring constant 860 N/m. The object is displaced from equilibrium 50.0 cm horizontally and given an initial velocity of 9.9 m/s back toward the equilibrium position.
(a) What is the frequency of the motion?
units in Hz
(b) What is the initial potential energy of the block-spring system?
units in J
(c) What is the initial kinetic energy?
units in J
(d) What is the amplitude of the motion?
units in m
(a) There is a standard formula for the frequency of spring/mass SHM.
f = [1/(2 pi)]*sqrt(k/M)
Use it.
(b) Ditto the P.E. of a stretched spring, (1/2) k X^2.
(c) (1/2) M Vo^2
(d) Amplitude X is whatever the P.E. is when K.E. = 0. That will happen when P.E. = total energy (b) + (c)
To answer these questions, we need to use the concepts of simple harmonic motion and energy conservation in the spring-block system. Let's break down each question and explain how to find the answers.
(a) Frequency (f) of the motion can be calculated using the formula:
f = (1 / (2π)) * √(k / m)
where k is the spring constant and m is the mass of the object.
First, convert the displacement of 50.0 cm to meters: 50.0 cm = 0.50 m.
Since the object is initially displaced and released with an initial velocity back towards the equilibrium position, we can assume that it performs simple harmonic motion. Thus, its mass (m) is relevant to the calculation of frequency.
Using the given values:
m = 5.80 kg
k = 860 N/m
Now, plug these values into the formula to find the frequency:
f = (1 / (2π)) * √(860 N/m / 5.80 kg)
Simplify the expression and calculate the square root to find the frequency in Hz.
(b) Initial Potential Energy (U) of the block-spring system can be calculated using the formula:
U = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
Using the given values:
k = 860 N/m
x = 0.50 m
Plug these values into the formula to find the initial potential energy in joules.
(c) Initial Kinetic Energy (K) of the block-spring system can be calculated by considering that energy is conserved:
K = (1/2) * m * v^2
where m is the mass of the object and v is the initial velocity.
Using the given values:
m = 5.80 kg
v = 9.9 m/s
Plug these values into the formula to find the initial kinetic energy in joules.
(d) Amplitude (A) of the motion is the maximum displacement of the object from the equilibrium position. In this case, it is the absolute value of the displacement:
A = |x|
where x is the displacement, as given in part (b).
Calculate the amplitude by taking the absolute value of the displacement.
By following these steps, you can find the answers to all the questions.