1. A 3 kg block collides with a massless spring of spring constant 90 N/m attached to a wall. The speed of the block was observed to be 1.5 m/s at the moment of collision. The acceleration of gravity is 9.8 m/s^2. How far does the spring compress if the surface on which the mass moves is frictionless?
Answer in units of cm.
2. The maximum distance to which the spring was compressed was observed to be 16.4317 cm. What is the kinetic coefficient of friction between the block and the floor?
3. Given: The coefficient of static friction between the floor and the block is 0.670681. Does the block remain at rest or does it
bounce back off the spring once the spring is fully compressed?
1.stays at rest
2.cannot be determined
3.bounces back
1. To solve this problem, we can use the principle of conservation of mechanical energy. The initial mechanical energy of the system (block-spring) is equal to the final mechanical energy.
The initial mechanical energy is the kinetic energy of the block just before the collision. The final mechanical energy is the potential energy stored in the compressed spring.
The kinetic energy (KE) of the block before the collision can be calculated using the formula:
KE = (1/2)mv^2
where m is the mass of the block (3 kg) and v is the speed of the block (1.5 m/s). Plugging in these values, we have:
KE = (1/2)(3 kg)(1.5 m/s)^2 = 3.375 J
The potential energy (PE) stored in the compressed spring can be calculated using the formula:
PE = (1/2)kx^2
where k is the spring constant (90 N/m) and x is the compression distance of the spring.
Now, equating the initial kinetic energy to the final potential energy, we have:
3.375 J = (1/2)(90 N/m)(x^2)
Simplifying the equation, we get:
x^2 = (2)(3.375 J)/(90 N/m) = 0.075 m^2
Finally, to find the compression distance in centimeters, we take the square root of 0.075 m^2 and convert it to centimeters:
x = sqrt(0.075 m^2) = 0.274 m ≈ 27.4 cm
Therefore, the spring compresses approximately 27.4 cm.
2. To find the kinetic coefficient of friction between the block and the floor, we can use the formula for maximum compression distance of a spring due to friction:
x = (u * m * g) / k
where x is the maximum compression distance (16.4317 cm), u is the kinetic coefficient of friction, m is the mass of the block (3 kg), g is the acceleration due to gravity (9.8 m/s^2), and k is the spring constant (90 N/m).
Rearranging the formula, we have:
u = (k * x) / (m * g)
Plugging in the given values, we have:
u = (90 N/m * 0.164317 m) / (3 kg * 9.8 m/s^2) ≈ 0.328
Therefore, the kinetic coefficient of friction between the block and the floor is approximately 0.328.
3. To determine whether the block remains at rest or bounces back off the spring once the spring is fully compressed, we need to compare the coefficient of static friction (μs) between the floor and the block with the kinetic coefficient of friction (u) that we calculated in question 2.
If the coefficient of static friction (μs) is greater than or equal to the kinetic coefficient of friction (u), the block will remain at rest after the spring is fully compressed. However, if μs is less than u, the block will bounce back.
Since the coefficient of static friction (μs) was not given in the problem statement, we cannot determine whether the block stays at rest or bounces back. The answer is option 2: cannot be determined.