Question 1

Two springs,each with spring constant k,are attached end to end in a series fashion.A second pair of identical springs are also attached in series fashion.Then the pair of series attached springs are attached in parallel to each other.What is the effective spring constant of the resulting combination of springs?

Accepted Answer

D

Question 2

A mass of 4.4 kg,resting on a horizontal frictionless surface,is attached on the right to a horizontal spring with spring constant 26 N/m and on the left to a horizontal spring with spring constant 58 N/m.If this system is moved from equilibrium,what is the effective spring constant?
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Accepted Answer

The effective spring constant of the system is the total force on the mass divided by the displacement.

Let x be the displacement of the mass from equilibrium.

The force exerted by the first spring is F1 = -k1x, where k1 is the spring constant of the first spring. 
The force exerted by the second spring is F2 = -k2x, where k2 is the spring constant of the second spring. 
The total force on the mass is therefore F = F1 + F2 = (-k1 - k2)x.

The effective spring constant of the system is Keff = F/x = -k1 - k2.

Plugging in the given values, we get

Keff = -26 N/m - 58 N/m = -84 N/m,

so the answer is (C).

Question 3

Two springs,each with spring constant 10 N/m,are connected in parallel,and the combination is hung vertically from one end.A 20-N weight is then suspended from the other end of the combined spring.How far does the combined spring stretch as it comes to equilibrium?
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Accepted Answer

The combined spring constant of the two springs in parallel can be found using the equation:
1/k = 1/k1 + 1/k2
where k1 and k2 are the spring constants of the individual springs. 
Plugging in the values, we get:
1/k = 1/10 + 1/10
1/k = 1/5
k = 5 N/m
Using Hooke's law, F = kx, we can find the displacement of the spring:
x = F/k = 20 N / 5 N/m = 4 m 
Therefore, the combined spring stretches 4 m when it comes to equilibrium.

Question 4

Which is not an example of approximate simple harmonic motion?

Accepted Answer

A ball bouncing on the floor does not exhibit approximate simple harmonic motion because the motion is not purely back-and-forth. Once the ball hits the floor, it bounces back up, but then experiences a brief moment of pause at the highest point of its bounce before falling back towards the ground again. This interruption in the motion breaks the criteria for simple harmonic motion.

Question 5

A 0.36-kg object is oscillating on a spring with a spring constant of k = 18 N/m.What is the potential energy of the system when the object displacement is 0.041 m,exactly half the maximum amplitude?
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Accepted Answer

At a displacement of half the maximum amplitude, the object has moved from its equilibrium position, thus it has potential energy stored in the spring. The potential energy of a spring is given by:

U = (1/2)kx^2

where U is the potential energy, k is the spring constant, and x is the displacement.

Plugging in the given values, we get:

U = (1/2)(18 N/m)(0.041 m)^2 
U = 0.015 J

Therefore, the correct answer is (C) 0.015 J.

Question 6

Three identical springs each have the same spring constant k.If these three springs are attached end to end forming a spring three times the length of one of the original springs,what will be the spring constant of the combination?

Accepted Answer

When three identical springs are attached end to end, they form a series connection. For a series connection of springs, the equivalent spring constant is given by:
1/keq = 1/k1 + 1/k2 + 1/k3
Since all three springs have the same spring constant k, the above formula simplifies to:
1/keq = 1/k + 1/k + 1/k = 3/k
Solving for keq, we get keq = k/3. Therefore, the spring constant of the combination is one-third of the original spring constant.

Question 7

A mass of 0.4 kg,hanging from a spring with a spring constant of 87 N/m,is set into an up-and-down simple harmonic motion.What is the speed of the mass when moving through a point at 0.05 m displacement? The starting displacement of the mass is 0.1 m from its equilibrium position.

Accepted Answer

The speed of the mass when moving through a point at 0.05 m displacement can be found using the equation:v = sqrt(k/m) * sqrt(A^2 - x^2)where:k is the spring constant (87 N/m)m is the mass (0.4 kg)A is the starting displacement (0.1 m)x is the displacement at which the speed is being calculated (0.05 m)Plugging in the values, we get:v = sqrt(87 N/m / 0.4 kg) * sqrt(0.1^2 m^2 - 0.05^2 m^2)v = 0.7 m/s

Question 8

A 0.25-kg object is attached to a spring with spring constant k = 10 N/m and moves with simple harmonic motion over a horizontal frictionless surface.At the instant that it is displaced from equilibrium by −0.051 m,what is its acceleration?
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Accepted Answer

The acceleration in simple harmonic motion is given by a = -kx/m. Substituting the given values: a = -(10 N/m)(-0.051 m) / 0.25 kg = 2 m/s².

Question 9

The SI base units for spring constant are which of the following?

Accepted Answer

The spring constant (k) has units of force per unit length (N/m). Since 1 N (Newton) is equivalent to 1 kg⋅m/s², the units for the spring constant are kg/s².

Question 10

Doubling the displacement from equilibrium of an object in simple harmonic motion will bring about a change in the magnitude of the object's acceleration by what factor?
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Accepted Answer

The magnitude of acceleration in simple harmonic motion is directly proportional to the displacement from equilibrium, so doubling the displacement will double the acceleration. Therefore, the factor of change is 2.

Question 11

Suppose there is an object for which F = +kx.What will happen if the object is moved away from equilibrium (x = 0)and released?

Accepted Answer

The answer of Suppose there is an object for which...

Question 12

A mass of 0.4 kg,attached to a spring with a spring constant of 80 N/m,is set into simple harmonic motion.What is the magnitude of the acceleration of the mass when at its maximum displacement of 0.2 m from the equilibrium position?
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Accepted Answer

The acceleration at maximum displacement is given by a = (k/m) * x, where k is the spring constant, m is the mass, and x is the displacement. Substituting the given values: a = (80 N/m / 0.4 kg) * 0.2 m = 40 m/s².

Question 13

Tripling the weight suspended vertically from a coil spring will result in a change in the displacement of the spring's lower end by what factor?
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Accepted Answer

According to Hooke's law, the displacement of the spring is directly proportional to the weight suspended. Therefore, if we triple the weight suspended (3 times the original weight), the displacement will also increase by a factor of 3. Hence, the answer is C.

Question 14

Spring #1 has spring constant 80 N/m.Spring #2 has an unknown spring constant,but when connected in series with Spring #1,the connected springs have an effective force constant of 10 N/m.What is the spring constant for Spring #2?
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Accepted Answer

When two springs are connected in series, their effective force constant is given by:

1/k = 1/k1 + 1/k2

where k1 and k2 are the individual force constants of the springs.

Substituting the given values, we get:

1/10 = 1/80 + 1/k2

Solving for k2, we get:

k2 = 11 N/m

Therefore, the answer is D - 11 N/m.

Question 15

A mass of 0.39 kg,hanging from a spring with a spring constant of 72 N/m,is set into an up-and-down simple harmonic motion.What is the speed of the mass when moving through the equilibrium point? The starting displacement from equilibrium is 0.14 m.
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Accepted Answer

The speed at the equilibrium point is the maximum speed, which can be calculated using the formula v_max = sqrt(k/m) * A, where k is the spring constant, m is the mass, and A is the amplitude. Substituting the given values, v_max = sqrt(72/0.39) * 0.14 = 1.9 m/s.

Question 16

If it takes 3 N to stretch a spring 6 cm and if the spring is then cut in half,what force does it take to stretch one of the halves 3 cm?
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Accepted Answer

The answer of If it takes 3 N to stretch...

Question 17

A 0.39 kg object,attached to a spring with spring constant k = 15 N/m,is moving on a horizontal frictionless surface in simple harmonic motion of amplitude of 0.082 m.What is its speed at the instant when its displacement is 0.041 m? (Hint: Use conservation of energy. )
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Accepted Answer

The conservation of energy principle states that the sum of kinetic and potential energy of a system remains constant, provided no external non-conservative forces act on the system.

At the equilibrium position (displacement = 0), all the potential energy is converted to kinetic energy, and at the displacement of 0.041 m, the object should have lost half of the potential energy at the equilibrium position.

Therefore, we can write:

(1/2) kx^2 = (1/2) mv^2

where k is the spring constant, x is the displacement, m is the mass, and v is the velocity.

Solving for v, we get:

v = sqrt((k/m) x^2)

Substituting the given values, we get:

v = sqrt((15/0.39) * (0.041)^2)

v = 0.44 m/s or 44 cm/s (rounded to one decimal place)

Therefore, the answer is C, 44 cm/s.

Question 18

A large spring requires a force of 190 N to compress it only 0.010 m.What is the spring constant of the spring?
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Accepted Answer

The spring constant, k, can be calculated using the formula: 
k = F/x 
where F is the force applied to compress the spring and x is the displacement. 
k = 190 N/0.01 m = 19,000 N/m. Therefore, the spring constant is 19,000 N/m, which corresponds to choice B.

Question 19

A runaway railroad car,with mass 36 × 10⁴ kg,coasts across a level track at 2.7 m/s when it collides with a spring-loaded bumper at the end of the track.If the spring constant of the bumper is 2.7 × 10⁶ N/m,what is the maximum compression of the spring during the collision? (Assume the collision is elastic. )

Accepted Answer

Explanation:The kinetic energy of the railroad car is converted to elastic potential energy in the spring. The maximum compression of the spring occurs when all of the kinetic energy has been converted to elastic potential energy. The kinetic energy of the railroad car is:$$KE = \frac{1}{2}mv^2 = \frac{1}{2}(36 	imes 10^4 	ext{ kg})(2.7 	ext{ m/s})^2 = 1.33 	imes 10^6 	ext{ J}$$The elastic potential energy stored in the spring is:$$PE = \frac{1}{2}kx^2$$where k is the spring constant and x is the compression of the spring. Setting the kinetic energy equal to the elastic potential energy, we get:$$1.33 	imes 10^6 	ext{ J} = \frac{1}{2}(2.7 	imes 10^6 	ext{ N/m})x^2$$Solving for x, we get:$$x = \sqrt{\frac{2(1.33 	imes 10^6 	ext{ J})}{2.7 	imes 10^6 	ext{ N/m}}} = 0.99 	ext{ m}$$Therefore, the maximum compression of the spring is 0.99 m.

Question 20

A tiny spring,with a spring constant of 1.3 N/m,will be stretched to what displacement by a 0.005-N force?
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Accepted Answer

The formula for the displacement caused by a force acting upon a spring is:

x = F/k

where:
x is the displacement of the spring
F is the force applied to the spring
k is the spring constant

Using the values given in the problem, we have:

x = (0.005 N)/(1.3 N/m)
x = 0.003846 m = 3.846 mm

Therefore, the best choice is A) 3.8 mm, since it is the closest option to the calculated value.

Quiz 13: Vibrations and Waves

A mass of 4.4 kg,resting on a horizontal frictionless surface,is attached on the right to a horizontal spring with spring constant 26 N/m and on the left to a horizontal spring with spring constant 58 N/m.If this system is moved from equilibrium,what is the effective spring constant?

Two springs,each with spring constant 10 N/m,are connected in parallel,and the combination is hung vertically from one end.A 20-N weight is then suspended from the other end of the combined spring.How far does the combined spring stretch as it comes to equilibrium?

Which is not an example of approximate simple harmonic motion?

A 0.36-kg object is oscillating on a spring with a spring constant of k = 18 N/m.What is the potential energy of the system when the object displacement is 0.041 m,exactly half the maximum amplitude?

Three identical springs each have the same spring constant k.If these three springs are attached end to end forming a spring three times the length of one of the original springs,what will be the spring constant of the combination?

A mass of 0.4 kg,hanging from a spring with a spring constant of 87 N/m,is set into an up-and-down simple harmonic motion.What is the speed of the mass when moving through a point at 0.05 m displacement? The starting displacement of the mass is 0.1 m from its equilibrium position.

A 0.25-kg object is attached to a spring with spring constant k = 10 N/m and moves with simple harmonic motion over a horizontal frictionless surface.At the instant that it is displaced from equilibrium by −0.051 m,what is its acceleration?

The SI base units for spring constant are which of the following?

Doubling the displacement from equilibrium of an object in simple harmonic motion will bring about a change in the magnitude of the object's acceleration by what factor?

Suppose there is an object for which F = +kx.What will happen if the object is moved away from equilibrium (x = 0) and released?

A mass of 0.4 kg,attached to a spring with a spring constant of 80 N/m,is set into simple harmonic motion.What is the magnitude of the acceleration of the mass when at its maximum displacement of 0.2 m from the equilibrium position?

Tripling the weight suspended vertically from a coil spring will result in a change in the displacement of the spring's lower end by what factor?

Spring #1 has spring constant 80 N/m.Spring #2 has an unknown spring constant,but when connected in series with Spring #1,the connected springs have an effective force constant of 10 N/m.What is the spring constant for Spring #2?

A mass of 0.39 kg,hanging from a spring with a spring constant of 72 N/m,is set into an up-and-down simple harmonic motion.What is the speed of the mass when moving through the equilibrium point? The starting displacement from equilibrium is 0.14 m.

If it takes 3 N to stretch a spring 6 cm and if the spring is then cut in half,what force does it take to stretch one of the halves 3 cm?

A 0.39 kg object,attached to a spring with spring constant k = 15 N/m,is moving on a horizontal frictionless surface in simple harmonic motion of amplitude of 0.082 m.What is its speed at the instant when its displacement is 0.041 m? (Hint: Use conservation of energy. )

A large spring requires a force of 190 N to compress it only 0.010 m.What is the spring constant of the spring?

A tiny spring,with a spring constant of 1.3 N/m,will be stretched to what displacement by a 0.005-N force?

Quiz 13: Vibrations and Waves

A mass of 4.4 kg,resting on a horizontal frictionless surface,is attached on the right to a horizontal spring with spring constant 26 N/m and on the left to a horizontal spring with spring constant 58 N/m.If this system is moved from equilibrium,what is the effective spring constant? ​

Two springs,each with spring constant 10 N/m,are connected in parallel,and the combination is hung vertically from one end.A 20-N weight is then suspended from the other end of the combined spring.How far does the combined spring stretch as it comes to equilibrium? ​

Which is not an example of approximate simple harmonic motion?

A 0.36-kg object is oscillating on a spring with a spring constant of k = 18 N/m.What is the potential energy of the system when the object displacement is 0.041 m,exactly half the maximum amplitude? ​

Three identical springs each have the same spring constant k.If these three springs are attached end to end forming a spring three times the length of one of the original springs,what will be the spring constant of the combination?

A mass of 0.4 kg,hanging from a spring with a spring constant of 87 N/m,is set into an up-and-down simple harmonic motion.What is the speed of the mass when moving through a point at 0.05 m displacement? The starting displacement of the mass is 0.1 m from its equilibrium position.

A 0.25-kg object is attached to a spring with spring constant k = 10 N/m and moves with simple harmonic motion over a horizontal frictionless surface.At the instant that it is displaced from equilibrium by −0.051 m,what is its acceleration? ​

The SI base units for spring constant are which of the following?

Doubling the displacement from equilibrium of an object in simple harmonic motion will bring about a change in the magnitude of the object's acceleration by what factor? ​

Suppose there is an object for which F = +kx.What will happen if the object is moved away from equilibrium (x = 0) and released?

A mass of 0.4 kg,attached to a spring with a spring constant of 80 N/m,is set into simple harmonic motion.What is the magnitude of the acceleration of the mass when at its maximum displacement of 0.2 m from the equilibrium position? ​

Tripling the weight suspended vertically from a coil spring will result in a change in the displacement of the spring's lower end by what factor? ​

Spring #1 has spring constant 80 N/m.Spring #2 has an unknown spring constant,but when connected in series with Spring #1,the connected springs have an effective force constant of 10 N/m.What is the spring constant for Spring #2? ​

A mass of 0.39 kg,hanging from a spring with a spring constant of 72 N/m,is set into an up-and-down simple harmonic motion.What is the speed of the mass when moving through the equilibrium point? The starting displacement from equilibrium is 0.14 m. ​

If it takes 3 N to stretch a spring 6 cm and if the spring is then cut in half,what force does it take to stretch one of the halves 3 cm? ​

A 0.39 kg object,attached to a spring with spring constant k = 15 N/m,is moving on a horizontal frictionless surface in simple harmonic motion of amplitude of 0.082 m.What is its speed at the instant when its displacement is 0.041 m? (Hint: Use conservation of energy. ) ​

A large spring requires a force of 190 N to compress it only 0.010 m.What is the spring constant of the spring? ​

A tiny spring,with a spring constant of 1.3 N/m,will be stretched to what displacement by a 0.005-N force? ​

A mass of 4.4 kg,resting on a horizontal frictionless surface,is attached on the right to a horizontal spring with spring constant 26 N/m and on the left to a horizontal spring with spring constant 58 N/m.If this system is moved from equilibrium,what is the effective spring constant?

Two springs,each with spring constant 10 N/m,are connected in parallel,and the combination is hung vertically from one end.A 20-N weight is then suspended from the other end of the combined spring.How far does the combined spring stretch as it comes to equilibrium?

A 0.36-kg object is oscillating on a spring with a spring constant of k = 18 N/m.What is the potential energy of the system when the object displacement is 0.041 m,exactly half the maximum amplitude?

A 0.25-kg object is attached to a spring with spring constant k = 10 N/m and moves with simple harmonic motion over a horizontal frictionless surface.At the instant that it is displaced from equilibrium by −0.051 m,what is its acceleration?

Doubling the displacement from equilibrium of an object in simple harmonic motion will bring about a change in the magnitude of the object's acceleration by what factor?

A mass of 0.4 kg,attached to a spring with a spring constant of 80 N/m,is set into simple harmonic motion.What is the magnitude of the acceleration of the mass when at its maximum displacement of 0.2 m from the equilibrium position?

Tripling the weight suspended vertically from a coil spring will result in a change in the displacement of the spring's lower end by what factor?

Spring #1 has spring constant 80 N/m.Spring #2 has an unknown spring constant,but when connected in series with Spring #1,the connected springs have an effective force constant of 10 N/m.What is the spring constant for Spring #2?

A mass of 0.39 kg,hanging from a spring with a spring constant of 72 N/m,is set into an up-and-down simple harmonic motion.What is the speed of the mass when moving through the equilibrium point? The starting displacement from equilibrium is 0.14 m.

If it takes 3 N to stretch a spring 6 cm and if the spring is then cut in half,what force does it take to stretch one of the halves 3 cm?

A 0.39 kg object,attached to a spring with spring constant k = 15 N/m,is moving on a horizontal frictionless surface in simple harmonic motion of amplitude of 0.082 m.What is its speed at the instant when its displacement is 0.041 m? (Hint: Use conservation of energy. )

A large spring requires a force of 190 N to compress it only 0.010 m.What is the spring constant of the spring?

A tiny spring,with a spring constant of 1.3 N/m,will be stretched to what displacement by a 0.005-N force?