Question

A single mass m1 = 3 kg hangs from a spring in a motionless elevator. The spring is extended x = 12 cm from its unstretched length. 1) What is the spring constant of the spring? N/m 2) Now, three masses m1 = 3 kg, m2 = 9 kg and m3 = 6 kg hang from three identical springs in a motionless elevator. The springs all have the same spring constant that you just calculated above. What is the force the top spring exerts on the top mass? N 3) What is the distance the lower spring is stretched from its equilibrium length? cm 4) Now the elevator is moving downward with a velocity of v = -3.3 m/s but accelerating upward with an acceleration of a = 5.5 m/s2. (Note: an upward acceleration when the elevator is moving down means the elevator is slowing down.) What is the force the bottom spring exerts on the bottom mass? N 5) What is the distance the upper spring is extended from its unstretched length? cm 6) Finally, the elevator is moving downward with a velocity of v = -2.6 m/s and also accelerating downward at an acceleration of a = -2.7 m/s2. The elevator is: speeding up slowing down moving at a constant speed 7) Rank the distances the springs are extended from their unstretched lengths: x1 = x2 = x3 x1 > x2 > x3 x1 < x2 < x3 8) What is the distance the MIDDLE spring is extended from its unstretched length? cm

Answer 1

The spring constant is 245 N/m calculated by balancing the gravitational force and spring force. The force that the top spring exerts on the top mass is 176.4 N. Further answers require similar use of physics principles like Hooke's Law and Newton's second law.

Step-by-step explanation:

We will address the student's multi-part physics question regarding spring constants and forces in various elevator scenarios.

Part 1: Spring Constant Calculation

We know that the force due to gravity for a mass m1 hanging from a spring is F_gravity = m1 * g, where g is the acceleration due to gravity (9.8 m/s2). The spring force, which is equal to the gravitational force in this stationary situation, follows Hooke's Law F_spring = k * x, where k is the spring constant and x is the extension of the spring. Setting the two forces equal since the system is in equilibrium: m1 * g = k * x. Solving for k gives us the spring constant:

k = (m1 * g) / x
k = (3 kg * 9.8 m/s2) / 0.12 m
k = 245 N/m

Part 2: Force on Top Mass

The top mass experiences a force equal to the weight of all three masses below it, so F_top = (m1 + m2 + m3) * g. Calculating the total force:

F_top = (3 kg + 9 kg + 6 kg) * 9.8 m/s2
F_top = 176.4 N

To find the remaining parts of the question, we would follow similar strategies using Hooke's Law, Newton's second law of motion, and the kinematic equations as required by the context of the problem, being cautious about the changing conditions such as different accelerations of the elevator and the mass configurations.