? asked in Science & MathematicsPhysics · 7 years ago

# An object with mass 2.5 kg is attached to a spring with spring stiffness constant k = 280 N/m and is executing?

An object with mass 2.5 kg is attached to a spring with spring stiffness constant k = 280 N/m and is executing simple harmonic motion. When the object is 0.020 m from its equilibrium position, it is moving with a speed of 0.55 m/s.

(a) Calculate the amplitude of the motion.

(b) Calculate the maximum velocity attained by the object. [Hint: Use conservation of energy.]

Relevance
• 7 years ago

The natural frequency, w, is w = sqrt(k/m)

w = sqrt(280 N/m / 2.5 kg) = 10.58 rad/s

Equations for SHM

x = A*sin(w*t)

v = w*A*cos(w*t)

a) Now, use the intial conditions to find the time and amplitude

0.020 m = A*sin(10.58 rad/s * t)

0.55 m/s = 10.58 rad/s * A * cos ( 10.58 rad/s * t)

Solve the upper equation for A and plug into the lower equation to solve for t. Once we have a number, then we can use either equation to solve for the value of A

A = 0.020 m / (sin(10.58 * t))

0.55 m/s = 10.58 rad/s * 0.020 m / (sin(10.58 * t)) * cos(10.58 rad/s * t)

0.55 m/s = 10.58 rad/s * 0.020 m * (cos (10.58 rad/s * t)) / (sin(10.58 rad/s * t)

0.55 m/s = 10.58 rad/s * 0.020 m * cot(10.58 rad/s * t)

cot(10.58 rad/s * t) = 2.60

cot = 1/tan so...

tan(10.58 rad/s * t ) = 1/2.60

10.58 rad/s * t = atan (1/2.60)

10.58 rad/s * t = 0.367

t = 0.367 / 10.58 rad/s = 0.0347 s

Now that we know time...

x = A*sin(w*t)

A = x / sin(w*t) = 0.020 m / sin(10.58 rad/s * 0.0347 s) = 0.0557 m

b) You could use conservation of energy, but I'll show you a quick way...

v = w*A*cos(w*t)

The maximum velocity occurs when the cos(w*t) term is equal to 1

v_max = w*A*1

v_max = w*A = 10.58 rad/s * 0.0557 m = 0.589 m/s

To show it with energy....

Max Spring Energy = 1/2*k*x^2, where x is the maximum displacement or amplitude of the motion.

That all gets transferred into Kinetic Energy

Max KE = Max Spring = 1/2*m*v^2

1/2*m*v^2 = 1/2*k*x^2

solve for v

v = sqrt(k/m*x^2)

v = x*sqrt(k/m)

Remember, sqrt(k/m) = w and x = amplitude

v = w*A

Just like I said, the max velocity occurs when the cos(w*t) term is equal to 1.