Physics 1120

Simple Harmonic Motion

1. A 1.75-kg particle moves as function of time as follows:
   x = 4cos(1.33t+φ/5)
   where distance is measured in metres and time in seconds.
   (a) What is the amplitude, frequency, angular frequency, and period of this motion?
   (b) What is the equation of the velocity of this particle?
   (c) What is the equation of the acceleration of this particle?
   (d) What is the spring constant?
   (e) At what next time t > 0, will the object be:
   
   1. at equilibrium and moving to the right,
   2. at equilibrium and moving to the left,
   3. at maximum amplitude, and
   4. at minimum amplitude.
2. If the amplitude in Question #1 is doubled, how would yours answers change?
3. What are the equations for the potential and kinetic energies of the particle in Question #1? What is the total energy?
4. The diagram below shows the motion of a 2.00-kg mass on a horizontal spring. Draw the reference circle. Find the phase constant. Write down the equation of the displacement as a function of time. What is the spring constant?What is the total energy? What is the maximum speed? What is the maximum acceleration? When exactly will the mass be at equilibrium and moving to the right? When exactly will the mass be at point C?

   

5. The diagram below shows the velocity of a 2.00-kg mass on a horizontal spring. What is the maximum amplitude of the object's displacement? What is the maximum acceleration? Draw the reference circle. What is the phase constant? Write down the equation of the displacement as a function of time. What is the spring constant? What is the total energy? When exactly will the mass have maximum positive velocity (point A)?

   

6. The diagram below shows the acceleration of a 2.00-kg mass on a horizontal spring. What is the maximum amplitude of the object's displacement?  What is the maximum velocity?  Draw the reference circle. Find the phase constant. Write down the equation of the displacement as a function of time.  What is the spring constant?  What is the total energy? When exactly will the mass first have minimum acceleration (point B)?

   

7. The diagram below shows a block oscillating back and forth at t = 0. Sketch the reference circle. Find the phase constant φ₀ in degrees. Sketch the x-t and v-t graphs.

   

8. The t = 0 reference circle shown is for a 4.0 kg block on a spring with stiffness K = 900 N/m. What is the phase constant φ₀? Sketch the x-t and v-t graphs. When exactly will the block reach equilibrium?

   

9. A horizontal spring with k = 200N/m has an attached mass of 0.150 kg. It is stretched and released. As the mass passes through the equilibrium point, its speed is 5.25 m/s. What was the amplitude of the motion?
10. A stiff spring k = 400 N/m has be attached to the floor vertically. A mass of 6.00 kg is placed on top of the spring as shown below and it finds a new equilibrium point. If the block is pressed downward and released it oscillates. If the compression is too big, however, the block will lose contact with the spring at the maximum vertical extension. Draw a free body diagram and find that extension at which the block loses contact with the spring.



11. A 100-g ball hangs from a metre-long string. Its swing is shown at t = 0 and it is moving to the right. Sketch the reference circle. Find the phase constant φ₀ in degrees. Sketch the x-t and v-t graphs. When will the ball pass through equilibrium?

    

12. In the diagram below, a mass on a string of length L encounters a nail positioned a distance L/n from the bottom of the string when the string hangs vertical. What is the period of this "interrupted" pendulum?



13. A block of mass M is on a frictionless surface as shown below.  It is attached to a wall by two springs with the same constant K.  Initially the block is at rest and the springs unstretched.  The block is pulled a distance A and then released.
    (a) What is the speed of the block as it passes through equilibrium?
    (b) What is the angular frequency ω of the motion?
    (c) If the two springs were replaced by one spring so that ω remains the same, what would its spring constant have to be?
    (d) If the two springs were kept, what would the mass have to be so that it has the same ω as a system comprised of a single block of mass M and spring with stiffness K?

    

14. A disk of mass M is on a surface as shown below.  It is attached to a wall by a spring of constant K.  Initially the disk is at rest and the spring is unstretched.  The disk is pulled a distance A and then released.
    (a) What is the speed ofthe block as it passes through equilibrium?
    (b) What is the angular frequency ω of the motion?
    (c) What would spring constant be for this system to have the same ω as a system comprised of a single block of mass M and spring with stiffness K?
    (d) What would the mass of the disk have to be for this system to have the same  ω  as a system comprised of a single block of mass M and spring with stiffness K?

    

15. A solid sphere of mass M is on a surface as shown below.  It is attached to a wall by two springs with the same constant K.  Initially the sphere is at rest and the springs unstretched.  The sphere is pulled a distance A and then released.
    (a) What is the speed of the sphere as it passes through equilibrium?
    (b) What is the angular frequency ω of the motion?
    (c) What would the spring constants have to be for this system to have the same ω as a system comprised of a single block of mass M and spring with stiffness K?
    (d) What would the mass of the sphere have to be for this system to have the same  ω  as a system comprised of a single block of mass M and spring with stiffness K?
    

    

16. A block with a mass M₁ = 5.00 kg is sliding to the right with a speed of 15.0 m/s when it hits and sticks to a block of mass, M₂ = 3.00 kg, attached to a spring of spring constant k = 3000 N/m. The second block was at the equilibrium position and the spring uncompressed at the time of the collision. The horizontal surface is frictionless.
    (a) Find the angular frequency of vibration for the system.
    (b) Find the maximum compression of the spring.
    (c) Find the maximum acceleration of the joined blocks.

    

Questions?mike.coombes@kwantlen.ca