At t = 0, a 785 g mass at rest on the end of a horizontal spring (k = 125 N/m) is struck by a hammer, which gives the mass an initial speed of 2.70 m/s.
(a) Determine the period of the motion.
T= .4979201367s
Determine the frequency of the motion.
f= 2.008354 Hz
(b) Determine the amplitude.
(c) Determine the maximum acceleration.
(d) Determine the position as a function of time.
(e) Determine the total energy.
m = 0.785 kg
k = 125 N/m
w = sqrt (k/m) = 2 pi f
T = 1/E at center of motion = total energy = (1/2) mv^2
KE = E = (1/2)(m )(2.70)^2
so
U at max swing amplitude A = E = (1/2) k A^2
so
A^2 = 2 E/k
max acceleration = w^2 A
x = A sin w t
we already did total energy because I chose to find amplitude that way.
To determine the amplitude, maximum acceleration, position as a function of time, and total energy, we need to use the equations of motion and applicable formulas. Here are the steps for each part:
(b) Determine the amplitude:
The amplitude of the motion can be determined using the formula for the maximum velocity reached by a mass-spring system: vmax = Aω, where A is the amplitude and ω is the angular frequency. We can determine ω using the formula ω = 2πf, where f is the frequency.
Step 1: Calculate the angular frequency:
ω = 2πf = 2π(2.008354 Hz)
Step 2: Calculate the amplitude:
A = vmax/ω = (2.70 m/s) / ω
(c) Determine the maximum acceleration:
The maximum acceleration of the mass can be found using the equation F = ma, where F is the force, m is the mass, and a is the acceleration. In a mass-spring system, the force is given by Hooke's law: F = -kx, where k is the spring constant and x is the displacement from the equilibrium position.
Step 1: Calculate the displacement at maximum acceleration:
The displacement at maximum acceleration is equal to the amplitude. Let's call it x_max.
Step 2: Calculate the maximum acceleration:
a_max = |F_max|/m = |-kx_max|/m = (kx_max)/m
(d) Determine the position as a function of time:
The position of the mass as a function of time can be determined using the equation x(t) = A * cos(ωt), where x(t) is the position at time t.
(e) Determine the total energy:
The total energy of the system can be determined using the equation E = (1/2)mv² + (1/2)kx², where E is the total energy, m is the mass, v is the velocity, k is the spring constant, and x is the displacement.
Step 1: Calculate the velocity at t = 0:
The velocity at t = 0 is given as 2.70 m/s.
Step 2: Calculate the displacement at t = 0:
The displacement at t = 0 is equal to the amplitude.
Step 3: Calculate the total energy:
E = (1/2)mv² + (1/2)kx² = (1/2)(m)(2.70 m/s)² + (1/2)(k)(x)^2
Now we can go through each step to find the answers.
To solve parts (b), (c), (d), and (e) of the problem, we need to find the equation of motion for the mass oscillating on the spring. We can start by finding the angular frequency (ω) of the motion using the formula:
ω = √(k/m)
where k is the spring constant and m is the mass. Plugging in the values provided:
k = 125 N/m
m = 0.785 kg
we can calculate ω:
ω = √(125 N/m / 0.785 kg)
Once we have ω, we can use it to solve the remaining parts of the problem. Let's go through each part step by step:
(b) Amplitude: The amplitude (A) is the maximum displacement from the equilibrium position. Since no information has been provided about it, we cannot determine it with the given data. Typically, the amplitude is given separately in problems like this.
(c) Maximum Acceleration: The maximum acceleration (A_max) can be determined using the formula:
A_max = ω^2 * A
where A is the amplitude. Since we don't have the amplitude, we cannot find the maximum acceleration.
(d) Position as a Function of Time: The position of the mass as a function of time can be described by a sinusoidal function. The general form is:
x(t) = A * cos(ωt + φ)
where x(t) is the position at time t, A is the amplitude, ω is the angular frequency, and φ is the phase constant.
To find φ, we need more information about the initial conditions. Specifically, we need to know the initial displacement and velocity of the mass. Without this information, we cannot determine the position as a function of time.
(e) Total Energy: The total energy (E_total) of the system is the sum of the potential energy (E_potential) and the kinetic energy (E_kinetic):
E_total = E_potential + E_kinetic
The potential energy of a spring is given by:
E_potential = (1/2)kA^2
where k is the spring constant and A is the amplitude.
The kinetic energy is given by:
E_kinetic = (1/2)mv^2
where m is the mass and v is the velocity.
To calculate the total energy, we need the amplitude (A) and the initial velocity (v), which are not provided in the given data. Therefore, we cannot determine the total energy of the system.