Question 1

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26s^-1)t]. What is the velocity of the mass when t = 0.815 s? A)- 13.3 cm/s B)13.3 cm/s C)8.02 cm/s D)- 8.02 cm/s E)0 cm/s

Accepted Answer

The velocity v(t) is the derivative of the position function x(t). v(t) = -12.3 cm * (1.26 s^-1) * sin[(1.26 s^-1)t]. Plugging t = 0.815 s gives v(0.815) = -13.3 cm/s.

Question 2

The tip of the needle of a sewing machine executes simple harmonic motion with an amplitude of 1.2 cm and an angular frequency ω = 62 rad/s. What is the total distance traveled by the needle in 1.9 s?
A)27 cm
B)40 cm
C)1.1 cm
D)90 cm
E)140 cm

Accepted Answer

The answer of The tip of the needle of a...

Question 3

The maximum displacement of a mass attached to a spring is 0.122 m, while its highest acceleration is 0.953 m/s². What is its maximum speed? A)0.358 m/s B)0.108 m/s C)0.831 m/s D)0.341 m/s E)0.538 m/s

Accepted Answer

The maximum speed (v_max) in simple harmonic motion is given by v_max = ωA, where ω is the angular frequency and A is the amplitude. The angular frequency can be found using the formula ω = sqrt(a_max/A), where a_max is the maximum acceleration. Substituting the given values, ω = sqrt(0.953/0.122), and then v_max = ω * 0.122, which calculates to approximately 0.341 m/s.

Question 4

What is the total energy of a 300-g mass that is attached to a horizontal spring with a force constant of 260 N/m and oscillates along a frictionless horizontal surface with an amplitude of 8.0 cm?
A)21 J
B)0.83 J
C)2.4 J
D)1.6 J
E)6.2 J

Accepted Answer

The total energy in a spring-mass system is given by the formula E = (1/2)kA^2, where k is the spring constant and A is the amplitude. Substituting the given values: E = (1/2)(260 N/m)(0.08 m)^2 = 0.83 J.

Question 5

A mass of 1.53 kg is attached to a spring and the system is undergoing simple harmonic oscillations with a frequency of 1.95 Hz and an amplitude of 7.50 cm. What is the speed of the mass when it is 3.00 cm from its equilibrium position?
A)0.0368 m/s
B)0.551 m/s
C)0.421 m/s
D)0.842 m/s
E)0.919 m/s

Accepted Answer

We can use the equation for the speed of an object undergoing simple harmonic motion at a given displacement from the equilibrium position:

v = ω√(A^2 - x^2)

where ω is the angular frequency (ω = 2πf) and x is the displacement from the equilibrium position.

Plugging in the given values, we get:

ω = 2πf = 2π(1.95 Hz) ≈ 12.24 rad/s
A = 7.50 cm = 0.075 m
x = 3.00 cm = 0.030 m

Thus, the speed of the mass at a displacement of 3.00 cm is:

v = ω√(A^2 - x^2) ≈ 0.842 m/s

Therefore, the correct answer is D.

Question 6

A mass of 0.150 kg is attached to a spring with a force constant of 3.58 N/m and undergoes simple harmonic oscillations with an amplitude of 7.50 cm. What is the total mechanical energy of the system?
A)0.0201 J
B)0.0101 J
C)0.269 J
D)0.134 J
E)0 J

Accepted Answer

The answer of A mass of 0.150 kg is attached...

Question 7

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26s^-1)t]. What is the acceleration of the mass when t = 0.825 s? A)9.89 cm/s² B)-9.89 cm/s² C)10.6 cm/s² D)-10.6 cm/s² E)0 cm/s²

Accepted Answer

The acceleration of the mass is given by the second derivative of the position function with respect to time. The position function is given by x = (12.3 cm) cos[(1.26s^-1)t]. The first derivative of the position function is given by v = dx/dt = -(12.3 cm)(1.26s^-1) sin[(1.26s^-1)t]. The second derivative of the position function is given by a = dv/dt = -(12.3 cm)(1.26s^-1)² cos[(1.26s^-1)t]. At t = 0.825 s, the acceleration of the mass is given by a = -(12.3 cm)(1.26s^-1)² cos[(1.26s^-1)(0.825 s)] = -9.89 cm/s².

Question 8

An air conditioning unit is vibrating in simple harmonic motion with a period of 0.27 s and a range (from the maximum in one direction to the maximum in the other) of 3.0 cm. At t = 0 it is at its central position and moving in the +x direction. What is its position when t = 55 s?
A)x = -0.430 cm
B)x = -0.511 cm
C)x = 0.511 cm
D)x = -1.41 cm
E)x = -1.44 cm

Accepted Answer

The answer of An air conditioning unit is vibrating in...

Question 9

A simple harmonic oscillator has an amplitude of 3.50 cm and a maximum speed of 28.0 cm/s. What is its speed when the displacement is 1.75 cm?
A)12.0 cm/s
B)24.2 cm/s
C)14.2 cm/s
D)15.0 cm/s
E)17.0 cm/s

Accepted Answer

The answer of A simple harmonic oscillator has an amplitude...

Question 10

A mass of 1.53 kg is attached to a spring and the system is undergoing simple harmonic oscillations with a frequency of 1.95 Hz and an amplitude of 7.50 cm. What is the total mechanical energy of the system?
A)0.844 J
B)0.646 J
C)0.633 J
D)0.955 J
E)0 J

Accepted Answer

The answer of A mass of 1.53 kg is attached...

Question 11

The amplitude of a 3.00-kg object in simple harmonic motion is 6.00 m. The maximum acceleration of the object is 5.00 m/s². What is the period of simple harmonic motion? A)6.88 s B)1.20 s C)0.278 s D)0.833 s E)7.54 s

Accepted Answer

The maximum acceleration in simple harmonic motion is given by $ a_{max} = \omega^2 A $, where $ \omega $ is the angular frequency and $ A $ is the amplitude. Solving for $ \omega $, we have $ \omega = \sqrt{\frac{a_{max}}{A}} = \sqrt{\frac{5.00}{6.00}} $. The period $ T $ is given by $ T = \frac{2\pi}{\omega} $. Calculating this gives approximately 6.88 s.

Question 12

The position of an air-track cart that is oscillating on a spring is given by x = (12.4 cm) cos[(6.35s^-1)t]. At what value of t after t = 0 s is the cart first located at x = 8.47 cm? A)4.34 s B)0.108 s C)0.129 s D)7.39 s E)7.75 s

Accepted Answer

We can set up an equation using the given information:
8.47 = 12.4cos[(6.35s^-1)t]
Divide both sides by 12.4:
0.682 = cos[(6.35s^-1)t]
Take the inverse cosine of both sides:
cos^-1(0.682) = (6.35s^-1)t
Solve for t:
t = cos^-1(0.682)/(6.35s^-1) = 0.129 s
Therefore, the correct choice is C.

Question 13

The position of a mass that is oscillating on a spring is given by x = (17.4 cm) cos[(5.46s^-1)t]. What is the period of this motion? A)1.74 s B)1.15 s C)1.32 s D)0.869 s E)0.314 s

Accepted Answer

The period of the motion is given by T = 2π/ω, where ω is the angular frequency. In this case, ω = 5.46s⁻¹, so T = 2π/5.46s⁻¹ ≈ 1.15 s. Therefore, the correct answer is (B).

Question 14

The velocity of a mass attached to a spring is given by v = (1.5 cm/s) sin(ωt + π/2), where ω = 3.0 rad/s. What is the corresponding expression for x?
A)x = -(4.50 cm) sin(ωt + π/2)
B)x = (4.50 cm) cos(ωt + π/2)
C)x = -(0.50 cm) cos(ωt - π/2)
D)x = -(0.50 cm) cos(ωt + π/2)
E)x = -(0.50 cm) sin(ωt + π/2)

Accepted Answer

The velocity is the derivative of position. Integrating v = (1.5 cm/s) sin(ωt + π/2) gives x = -(0.50 cm) cos(ωt + π/2).

Question 15

A 30-g bullet traveling horizontally at 300 m/s strikes a 1.0-kg block which is attached to a horizontal spring with a force constant of 2000 N/m and rests on a frictionless horizontal surface. The spring is on the far side of the block and aligned with the direction of travel of the bullet. The bullet becomes embedded in the block and, as a result of the impact, and the block slides against the spring. How far is the spring compressed before it reverses its direction of travel?
A)0.20 m
B)0.58 m
C)1.2 m
D)0.40 m
E)0.039 m

Accepted Answer

Momentum is conserved in the collision, so the block and bullet move off together with a speed of 27 m/s. The kinetic energy of the system is then:$$K=\frac{1}{2}(1.03	ext{ kg})(27	ext{ m/s})^2=365	ext{ J}$$This energy is stored in the spring when it is compressed a distance x:$$U=\frac{1}{2}kx^2=365	ext{ J}$$Solving for x:$$x=\sqrt{\frac{2U}{k}}=\sqrt{\frac{2(365	ext{ J})}{2000	ext{ N/m}}}=0.20	ext{ m}$$

Question 16

What is the maximum kinetic energy of a 2.00-kg mass moving with simple harmonic motion with an amplitude 5.00 cm and period 0.300 s?
A)1.10 J
B)0.444 J
C)0.222 J
D)0.174 J
E)0.0278 J

Accepted Answer

The maximum kinetic energy of a mass in simple harmonic motion is given by:$$K_{max}=\frac{1}{2}kA^2$$where k is the spring constant and A is the amplitude. The spring constant is given by:$$k=\frac{4\pi^2m}{T^2}$$where m is the mass and T is the period. Substituting these expressions into the equation for the maximum kinetic energy, we get:$$K_{max}=\frac{1}{2}\left(\frac{4\pi^2m}{T^2}ight)A^2$$Plugging in the given values, we get:$$K_{max}=\frac{1}{2}\left(\frac{4\pi^2(2.00	ext{ kg})}{(0.300	ext{ s})^2}ight)(0.0500	ext{ m})^2=1.10	ext{ J}$$

Question 17

The position of a mass that is oscillating on a spring is given by x = (18.3 cm) cos[(2.35s^-1)t]. What is the frequency of this motion? A)0.426 Hz B)1.34 Hz C)2.67 Hz D)0.128 Hz E)0.374 Hz

Accepted Answer

The answer of The position of a mass that is...

Question 18

A 4.20-g bullet embeds itself in a 20.0-kg block, which is attached to a horizontal spring with a force constant of 980 N/m. The maximum compression of the spring is 2.40 cm. What is the speed of the bullet just before it hits the block?
A)800 m/s
B)700 m/s
C)672 m/s
D)900 m/s
E)952 m/s

Accepted Answer

The answer of A 4.20-g bullet embeds itself in a...

Question 19

The maximum speed of a 3.00-kg object in simple harmonic motion is 4.00 m/s. The maximum acceleration of the object is 5.00 m/s². What is the period of simple harmonic motion? A)1.25 s B)0.417 s C)0.800 s D)5.03 s E)3.75 s

Accepted Answer

The answer of The maximum speed of a 3.00-kg object...

Question 20

A 16.2-g bullet with an initial speed of 850 m/s embeds itself in a 40.0-kg block, which is attached to a horizontal spring with a force constant of 1000 N/m. What is the maximum compression of the spring?
A)6.34 cm
B)5.57 cm
C)3.42 cm
D)1.26 cm
E)6.88 cm

Accepted Answer

The answer of A 16.2-g bullet with an initial speed...

Quiz 14: Oscillations

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26s^-1) t]. What is the velocity of the mass when t = 0.815 s?

The tip of the needle of a sewing machine executes simple harmonic motion with an amplitude of 1.2 cm and an angular frequency ω = 62 rad/s. What is the total distance traveled by the needle in 1.9 s?

The maximum displacement of a mass attached to a spring is 0.122 m, while its highest acceleration is 0.953 m/s². What is its maximum speed?

What is the total energy of a 300-g mass that is attached to a horizontal spring with a force constant of 260 N/m and oscillates along a frictionless horizontal surface with an amplitude of 8.0 cm?

A mass of 1.53 kg is attached to a spring and the system is undergoing simple harmonic oscillations with a frequency of 1.95 Hz and an amplitude of 7.50 cm. What is the speed of the mass when it is 3.00 cm from its equilibrium position?

A mass of 0.150 kg is attached to a spring with a force constant of 3.58 N/m and undergoes simple harmonic oscillations with an amplitude of 7.50 cm. What is the total mechanical energy of the system?

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26s^-1) t]. What is the acceleration of the mass when t = 0.825 s?

An air conditioning unit is vibrating in simple harmonic motion with a period of 0.27 s and a range (from the maximum in one direction to the maximum in the other) of 3.0 cm. At t = 0 it is at its central position and moving in the +x direction. What is its position when t = 55 s?

A simple harmonic oscillator has an amplitude of 3.50 cm and a maximum speed of 28.0 cm/s. What is its speed when the displacement is 1.75 cm?

A mass of 1.53 kg is attached to a spring and the system is undergoing simple harmonic oscillations with a frequency of 1.95 Hz and an amplitude of 7.50 cm. What is the total mechanical energy of the system?

The amplitude of a 3.00-kg object in simple harmonic motion is 6.00 m. The maximum acceleration of the object is 5.00 m/s². What is the period of simple harmonic motion?

The position of an air-track cart that is oscillating on a spring is given by x = (12.4 cm) cos[(6.35s^-1) t]. At what value of t after t = 0 s is the cart first located at x = 8.47 cm?

The position of a mass that is oscillating on a spring is given by x = (17.4 cm) cos[(5.46s^-1) t]. What is the period of this motion?

The velocity of a mass attached to a spring is given by v = (1.5 cm/s) sin(ωt + π/2) , where ω = 3.0 rad/s. What is the corresponding expression for x?

What is the maximum kinetic energy of a 2.00-kg mass moving with simple harmonic motion with an amplitude 5.00 cm and period 0.300 s?

The position of a mass that is oscillating on a spring is given by x = (18.3 cm) cos[(2.35s^-1) t]. What is the frequency of this motion?

A 4.20-g bullet embeds itself in a 20.0-kg block, which is attached to a horizontal spring with a force constant of 980 N/m. The maximum compression of the spring is 2.40 cm. What is the speed of the bullet just before it hits the block?

The maximum speed of a 3.00-kg object in simple harmonic motion is 4.00 m/s. The maximum acceleration of the object is 5.00 m/s². What is the period of simple harmonic motion?

A 16.2-g bullet with an initial speed of 850 m/s embeds itself in a 40.0-kg block, which is attached to a horizontal spring with a force constant of 1000 N/m. What is the maximum compression of the spring?