Answer the following questions for a mass that is hanging on a spring and oscillating up and down with simple harmonic motion. Note: the oscillation is small enough that the spring stays stretched beyond its rest length the entire time.
Options for each question are (Top, Bottom, Top and Bottom, Equilibrium, or Nowhere)
Where in the motion is the speed zero?
Where in the motion is the magnitude of the force from the spring on the mass zero?
Where in the motion is the magnitude of the net force on the mass zero?
Where in the motion is the speed a maximum?
Where in the motion is the magnitude of the force from the spring on the mass a maximum?
Where in the motion is the magnitude of the acceleration a maximum?
Where in the motion is the magnitude of the net force on the mass a maximum?
Where in the motion is the acceleration zero?
These last two are just yes or no.
When the object is at half its amplitude from equilibrium, is the magnitude of its acceleration at half its maximum value?
When the object is at half its amplitude from equilibrium, is its speed half its maximum speed?
A vibration platform oscillates up and down with an amplitude of 10.2 cm at a controlled variable frequency. Suppose a small rock of unknown mass is placed on the platform. At what frequency will the rock just begin to leave the surface so that it starts to clatter?
The answers to the questions are as follows:
1. Where in the motion is the speed zero? - Equilibrium
2. Where in the motion is the magnitude of the force from the spring on the mass zero? - Top and Bottom
3. Where in the motion is the magnitude of the net force on the mass zero? - Equilibrium
4. Where in the motion is the speed a maximum? - Equilibrium
5. Where in the motion is the magnitude of the force from the spring on the mass a maximum? - Top and Bottom
6. Where in the motion is the magnitude of the acceleration a maximum? - Top and Bottom
7. Where in the motion is the magnitude of the net force on the mass a maximum? - Top and Bottom
8. Where in the motion is the acceleration zero? - Equilibrium
Yes or No answers:
9. When the object is at half its amplitude from equilibrium, is the magnitude of its acceleration at half its maximum value? - Yes
10. When the object is at half its amplitude from equilibrium, is its speed half its maximum speed? - No
To answer these questions, let's first understand the motion of a mass hanging on a spring with simple harmonic motion.
In this system, the mass oscillates up and down about an equilibrium position. The force exerted by the spring on the mass is proportional to the displacement from the equilibrium position according to Hooke's Law. The resulting motion is sinusoidal, meaning that it can be described by a sine or cosine function.
Now, let's address each question:
1. Where in the motion is the speed zero?
The speed of the mass is zero at its maximum displacement from the equilibrium position, both at the top and bottom.
2. Where in the motion is the magnitude of the force from the spring on the mass zero?
At the equilibrium position, the magnitude of the force from the spring on the mass is zero. This occurs when the spring is neither stretched nor compressed.
3. Where in the motion is the magnitude of the net force on the mass zero?
The magnitude of the net force on the mass is zero at both the top and bottom of the motion. This is because the net force is the sum of the forces from the spring and gravity, and at these points, they cancel each other out.
4. Where in the motion is the speed a maximum?
The speed of the mass is a maximum at the equilibrium position (i.e., where it passes through the center or equilibrium point). This is because the spring force is at its maximum at this point, causing the mass to accelerate and attain its maximum speed.
5. Where in the motion is the magnitude of the force from the spring on the mass a maximum?
The magnitude of the force from the spring on the mass is a maximum at the maximum displacement from the equilibrium position (i.e., at the top and bottom of the motion). This is because the displacement from the equilibrium directly affects the magnitude of the restoring force from the spring.
6. Where in the motion is the magnitude of the acceleration a maximum?
The magnitude of the acceleration is a maximum at the maximum displacement from the equilibrium position (i.e., at the top and bottom of the motion). This is because the acceleration is directly proportional to the net force acting on the mass, which is at its maximum at these points.
7. Where in the motion is the magnitude of the net force on the mass a maximum?
Similar to the previous question, the magnitude of the net force on the mass is a maximum at the maximum displacement from the equilibrium position (i.e., at the top and bottom of the motion).
8. Where in the motion is the acceleration zero?
The acceleration is zero at the equilibrium position (i.e., when the mass is at the center or equilibrium point) since there is no net force acting on it at this point.
Now, let's handle the last two questions:
- When the object is at half its amplitude from equilibrium, is the magnitude of its acceleration at half its maximum value?
No, the magnitude of the acceleration is not at half its maximum value. The acceleration is directly proportional to the displacement, not the distance from the equilibrium. Hence, the acceleration is still at its maximum value at this point.
- When the object is at half its amplitude from equilibrium, is its speed half its maximum speed?
No, the speed is not half its maximum speed. The speed is determined by the position, and it varies sinusoidally with time. When the object is halfway between the equilibrium point and its maximum displacement, its speed is not necessarily half the maximum speed.