A mass of 3.4kg sliding along a frictionless floor at 1.5 m/s strikes and compresses a spring of constant k=295 N/m. The spring stops the mass. How far does the mass travel while being slowed by the spring? How long does the mass take to stop?
To find out how far the mass travels while being slowed by the spring, we can use the conservation of mechanical energy.
The initial kinetic energy of the mass can be determined using the formula: KE = (1/2)mv^2, where m is the mass and v is the velocity. Plugging in the given values, we get:
KE = (1/2)(3.4 kg)(1.5 m/s)^2 = 3.825 Joules
Since the spring stops the mass, all of the initial kinetic energy is transferred to potential energy stored in the spring. The potential energy stored in a spring can be calculated using the formula: PE = (1/2)kx^2, where k is the spring constant and x is the distance the spring is compressed. Rearranging this equation, we get:
x = √(2PE / k)
Plugging in the values for the potential energy and spring constant, we get:
x = √(2(3.825 J) / 295 N/m) ≈ 0.193 meters
Therefore, the mass travels approximately 0.193 meters while being slowed by the spring.
To determine how long the mass takes to stop, we can use the equation of motion for an object slowing down under the influence of a force. The force exerted by the spring can be calculated using Hooke's Law: F = kx, where F is the force, k is the spring constant, and x is the distance the spring is compressed. Rearranging this equation, we get:
x = F / k
Plugging in the values for the force and spring constant, we get:
x = 3.4 kg * 9.8 m/s^2 / 295 N/m ≈ 0.113 meters
So, the distance the mass travels while being slowed by the spring is approximately 0.113 meters.
To find the time it takes for the mass to stop, we can use the equation of motion:
v = u + at
Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. In this case, the final velocity is 0 m/s (since the mass stops) and the initial velocity is 1.5 m/s. The acceleration can be calculated using Newton's second law: F = ma. Rearranging the equation, we get:
a = F / m
Plugging in the values for the force and mass, we get:
a = 3.4 kg * 9.8 m/s^2 / 3.4 kg = 9.8 m/s^2
Now, we can rearrange the equation of motion to solve for time:
t = (v - u) / a = (0 m/s - 1.5 m/s) / (-9.8 m/s^2) ≈ 0.153 seconds
Therefore, it takes approximately 0.153 seconds for the mass to stop.