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In this lesson, we will learn how to calculate the instantaneous position, velocity, acceleration, and energy of simple harmonic spring oscillators.

Q1:

A mass π 0 is attached to a spring and hung vertically. The mass is raised a short distance in the vertical direction and released. The mass oscillates with a frequency π 0 . If the mass is replaced with a mass nine times as large, and the experiment was repeated, what would be the frequency of the oscillations in terms of π 0 ?

Q2:

Near the top of the Citigroup Center building in New York City, there is an object with mass of 4 . 0 0 Γ 1 0 5 kg on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being drivenβthe driving force is transferred to the object, which oscillates instead of the entire building.

What effective force constant should the springs have to make the object oscillate with a period of 2.00 s?

What energy is stored in the springs for a 2.00-m displacement from equilibrium?

Q3:

A type of clock keeps time by the oscillation of a small object bouncing on a spring. What force constant of a spring is needed to produce a period of 0.370 s for an object of mass 0.0210 kg?

Q4:

A block with a mass of 2.0 kg lies at rest on a frictionless table. A spring, with a spring constant of 1 . 0 Γ 1 0 2 N/m is attached to a wall at one end of the table, and the other end of the spring is attached to the block. A second block, of mass 0.50 kg, is placed on top of the first block, the centres of the blocks aligned with each other. The 2.0-kg-mass block is gently pulled away from the wall, to a position π₯ , and released from rest, after which the blocks oscillate on the end of the spring. There is a coefficient of friction of 0.45 between the two blocks as they oscillate.

What is the period of the oscillations of the system of blocks?

What is the largest value of initial spring extension π₯ for which the centres of the two boxes remain aligned at all times during the boxes oscillation?

Q5:

A block of mass 200 g is attached at the end of a massless spring of spring constant 100 N/cm. The other end of the spring is attached to the ceiling and the mass is brought to rest at a point π . Point π is taken to be the zero of the potential energy of the block, both from the weight and the spring force. The mass hangs freely and the spring is in a stretched state. The block is then pulled downward by another 5.00 cm and released from rest.

What is the net potential energy of the block at the instant the block is at the lowest point?

What is the net potential energy of the block at the instant the block returns to the point marked π ?

What is the speed of the block as it crosses the point marked π ?

How high above the point marked π does the block rise before coming to rest again?

Q6:

Consider a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction π = 0 . 1 0 0 π .

How far can the spring be stretched without moving the mass?

The mass is set into oscillations that have an amplitude equal to twice the maximum distance that the spring can stretch without moving the object. During these oscillations, the kinetic coefficient of friction π = 0 . 8 5 0 π . If the object starts oscillating with maximum initial displacement, what total distance will it move before it comes to rest?

Q7:

The length of nylon rope from which a mountain climber is suspended has an effective force constant of 1 . 4 0 Γ 1 0 4 N/m.

The climber oscillates on the end of the rope. What is the frequency at which he bounces, given his mass plus the mass of his equipment is 90.0 kg?

How much will this rope stretch to break the climberβs fall if he free-falls 2.00 m before the rope runs out of slack?

How much will this rope stretch to break the climberβs fall if he free-falls 2.00 m before the rope runs out of slack and the effective force constant of the rope is 7 . 0 0 Γ 1 0 3 N/m?

Q8:

In a novelty clock, an object of mass 0.0100 kg is suspended from a spring with a force constant of 1.25 N/m. The object bounces, and is displaced a maximum of 3.00 cm from its equilibrium position as it bounces.

What is the maximum speed of the object?

What is the maximum kinetic energy of the object?

Q9:

Consider a block of mass 0.200 kg attached to a spring of spring constant 100 N/m. The block is placed on a frictionless table, and the other end of the spring is attached to the wall so that the spring is level with the table. The block is then pushed in so that the spring is compressed by 10.0 cm.

Find the speed of the block as it crosses the point where the spring is not stretched.

Find the speed of the block as it crosses the point 5.00 cm to the left of the point where the spring is not stretched.

Find the speed of the block as it crosses the point 5.00 cm to the right of the point where the spring is not stretched.

Q10:

A spring with a spring constant of 127 N/m, which can be stretched or compressed, is placed on a frictionless horizontal table. An object of mass 9.77 kg is attached to one end of the spring and the other end is anchored to a wall at one end of the table. The equilibrium position of the object is marked as zero. A student moves the object 6.2 cm from its equilibrium position, extending the spring, and then releases the mass. Consider displacement in the direction that extends the spring to be positive-valued.

Find the position of the object at π‘ = 4 . 0 s.

Find the velocity of the object at π‘ = 4 . 0 s.

Find the acceleration of the object at π‘ = 4 . 0 s.

Q11:

An object with a mass of 0.320 kg is attached to a vertical spring. Initially, the object is held at rest. The object is then released from rest at a point that corresponds to the position of the free end of the spring when unloaded. The amplitude of the oscillations that occur when the object is released is equal to the distance between the unloaded equilibrium length of the spring and its loaded equilibrium length. The spring has a force constant of 17.6 N/m.

The position of the free end of the unloaded spring is marked as π¦ = 0 . 0 0 m. What is the magnitude of the vertical displacement of the same end of the spring if the object is suspended from it?

If the object is set in motion by release from rest with the object at the position of the free end of the unloaded spring, find the amplitude of the oscillations that occur after the object is released.

If the object is set in motion by release from rest with the object at the position of the free end of the unloaded spring, find the magnitude of the objectβs maximum velocity while oscillating.

Find the magnitude of the force that the spring exerts on the mass when it is at its lowest point.

Q12:

An object of mass 0.792 kg is suspended from a spring. The object oscillates with a period of 1.3 s. By how much would the object need to increase its mass to change the oscillation period to 3.33 s?

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