Question

3. BONUS: A 2.0 kg mass is attached to a spring that is hooked to a wall. The mass undergoes horizontal oscillations between 25 to 65 cm from the wall. The spring constant is 32 N/m. The spring is compressed and then released so it is at maximum displacement when ? = 0. Hint: Use the information to write equations for the x-position, x-velocity, and x-acceleration as a function of time first and then evaluate at the time given.

a. What is the x-position at ? = 1.0 ??

b. What is the x-velocity at ? = 1.0 ??

c. What is the x-acceleration at ? = ?. 2 ??

d. What is the total mechanical energy?

Answer 1

here,

the mass of block , m = 2 kg

the amplitude of motion , A = (65 - 25) /2 cm = 20 cm

the spring constant , K = 32 N/m

the angular velocity , w = sqrt(K/m)

w = sqrt(32/2) = 4 rad/s

the equation of motion , x(t) = (65 - 25) + (- A * cos(w*t))

x(t) = (65 - 25) + (- 20 cm * cos(4t))

x(t) = 45 cm + (- 20 cm * cos(4t))

a)

at t = 1 s

the position , x(1) = 45 cm + (- 20 cm * cos(4 * 1))

x(1) = 58.04 cm

b)

the velocity , v(t) = dx(t)/dt

v(t) = 20 * 4 cm/s * sin(4t)

at t = 1 s

v(1) = 20 * 4 cm/s * sin(4*1)

v(1) = - 60.65 cm/s

c)

the x-acceleration , a(t) = dv(t)/dt

a(t) = 20 * 4^2 * cos(4t)

at t = 1 s

a(1) = 20 * 4^2 cm/s^2 * cos(4 * 1)

a(1) = -209 cm/s^2 = - 2.09 m/s^2

d)

the total mechanical energy , ME = 0.5 *K * A^2

ME = 0.5 * 32 * 0.2^2 J

ME = 0.64 J