Work, Energy, and Power Practice Test

1.

A block of mass m slides down a rough incline of angle θ and height h. It loses 20% of its mechanical energy to friction by the time it reaches the bottom. What is the speed of the block at the bottom?

a) √(2gh) b) √(1.6gh) c) √(0.8gh) d) √(2gh cosθ)

2.

A particle moves under the influence of a position-dependent force F(x) = -kx + αx³, where k and α are constants. Which of the following expressions represents the potential energy function U(x) (assuming U(0) = 0)?

a) \( U(x) = \frac{1}{2}kx^2 + \frac{1}{4}\alpha x^4 \) b) \( U(x) = -\frac{1}{2}kx^2 + \frac{1}{4}\alpha x^4 \) c) \( U(x) = \frac{1}{2}kx^2 - \frac{1}{4}\alpha x^4 \) d) \( U(x) = -\frac{1}{2}kx^2 - \frac{1}{4}\alpha x^4 \)

3.

A spring (with spring constant k) is compressed by a distance x and used to launch a mass m up a frictionless incline angled at θ. What is the maximum distance d the mass travels along the incline before momentarily coming to rest?

a) \( d = \frac{kx^2}{2mg\sinθ} \) b) \( d = \frac{kx^2}{mg\cosθ} \) c) \( d = \frac{kx^2 \sinθ}{2mg} \) d) \( d = \frac{kx^2}{mg} \)

4.

A block of mass m is attached to a spring and oscillates with amplitude A on a horizontal frictionless surface. At what position x (relative to equilibrium) does the kinetic energy equal the potential energy?

a) \( x = \frac{A}{\sqrt{2}} \) b) \( x = \frac{A}{2} \) c) \( x = \frac{A}{\sqrt{3}} \) d) \( x = \frac{A}{3} \)

5.

A 2-kg object is moving at 3 m/s on a rough horizontal surface. A constant force of 5 N is applied in the direction of motion for 4 seconds. If the coefficient of kinetic friction is 0.3, what is the final speed of the object?

a) 13.4 m/s b) 11.2 m/s c) 10.0 m/s d) 7.6 m/s

6.

A force \( F(x) = ax^2 + bx \) acts on a particle moving along the x-axis, where \( a \) and \( b \) are constants. What is the work done by this force as the particle moves from \( x = 0 \) to \( x = L \)?

a) \( aL^3 + bL^2 \) b) \( \frac{aL^3}{3} + \frac{bL^2}{2} \) c) \( \frac{aL^2}{2} + \frac{bL}{2} \) d) \( \frac{aL^2}{2} + bL \)

7.

A particle of mass m is moving in a circular path under the influence of a conservative central force such that the potential energy is given by \( U(r) = -\frac{k}{r} + \frac{λ}{r^2} \), where \( k \) and \( λ \) are positive constants. At equilibrium radius \( r_0 \), the net radial force is zero. Which expression gives \( r_0 \)?

a) \( r_0 = \frac{2λ}{k} \) b) \( r_0 = \frac{k}{λ} \) c) \( r_0 = \sqrt{\frac{2λ}{k}} \) d) \( r_0 = \frac{λ}{2k} \)

8.

A pendulum of length L is released from rest at an angle \( \theta_0 \). Assuming no friction, what is the speed of the pendulum bob at the lowest point of the swing?

a) \( v = \sqrt{2gL(1 - \cos\theta_0)} \) b) \( v = \sqrt{gL\cos\theta_0} \) c) \( v = \sqrt{2gL\cos\theta_0} \) d) \( v = \sqrt{gL(1 - \cos\theta_0)} \)

9.

An object is acted upon by a non-conservative force such that its total mechanical energy changes over time as \( E(t) = E_0 e^{-\gamma t} \). Which of the following best describes the power delivered by the non-conservative force at time t?

a) \( P(t) = -\gamma E_0 e^{-\gamma t} \) b) \( P(t) = \gamma E_0 e^{-\gamma t} \) c) \( P(t) = -\frac{dE}{dt} = \gamma^2 E_0 e^{-\gamma t} \) d) \( P(t) = -\gamma^2 t E_0 e^{-\gamma t} \)

10.

A uniform rope of mass m and length L is lifted vertically at constant speed from a horizontal surface. What is the work done to lift the entire rope to a height L above the surface?

a) \( W = mgL \) b) \( W = \frac{1}{2}mgL \) c) \( W = 2mgL \) d) \( W = \frac{1}{4}mgL \)

11.

A block slides down a frictionless incline of angle \( \theta \) and length \( L \). The block starts from rest. What is the work done by gravity on the block during its descent?

a) \( mgL \cos \theta \) b) \( mgL \sin \theta \) c) \( mgL \) d) \( mgL \tan \theta \)

12.

A spring with spring constant \( k \) is compressed by a distance \( x \) and then releases a block of mass \( m \) on a frictionless surface. If the block encounters a rough patch of length \( d \) with coefficient of kinetic friction \( \mu_k \), what is the speed of the block after crossing the rough patch?

a) \( \sqrt{\frac{k}{m}x^2 - 2\mu_k g d} \) b) \( \sqrt{\frac{k}{m}x^2 - \frac{2\mu_k g d}{m}} \) c) \( \sqrt{\frac{k}{m}x^2 - 2\mu_k m g d} \) d) \( \sqrt{\frac{k}{m}x^2 + 2\mu_k g d} \)

13.

An object of mass \( m \) moves under a velocity-dependent resistive force \( \vec{F} = -b \vec{v} \). The object starts with initial kinetic energy \( K_0 \). How much work does the resistive force do after the object slows to half its initial speed?

a) \( -\frac{3}{4} K_0 \) b) \( -\frac{1}{4} K_0 \) c) \( -\frac{1}{2} K_0 \) d) \( -\frac{7}{8} K_0 \)

14.

A particle moves along the x-axis with total mechanical energy \( E \) under a potential \( U(x) = U_0 \sin^2 \left(\frac{\pi x}{L}\right) \). What is the maximum kinetic energy of the particle?

a) \( E + U_0 \) b) \( E - U_0 \) c) \( E \) d) \( U_0 - E \)

15.

A particle of mass \( m \) moves under a force \( F(x) = -kx + bx^3 \), where \( k \) and \( b \) are positive constants. Near the equilibrium at \( x=0 \), which statement about the work done by the force for small oscillations is correct?

a) The work done is purely harmonic and proportional to \( x^2 \). b) The cubic term \( bx^3 \) does no work. c) The work includes anharmonic contributions and deviates from simple harmonic motion. d) The force is conservative only if \( b = 0 \).

16.

A block of mass \( m \) is attached to a spring (spring constant \( k \)) on a frictionless surface. The block is pulled from equilibrium and released. If a constant non-conservative force \( F \) acts opposite to the block's motion during the first half of the oscillation, what happens to the mechanical energy of the system after one full oscillation?

a) It remains the same. b) It decreases by \( 2 F A \), where \( A \) is amplitude. c) It decreases by \( F A \). d) It increases by \( F A \).

17.

A 2 kg block slides down a 30° frictional incline with coefficient of kinetic friction \( \mu_k = 0.2 \). If the block travels 5 m, what is the work done by friction?

a) \( -9.8 \, \text{J} \) b) \( -19.6 \, \text{J} \) c) \( -16.96 \, \text{J} \) d) \( -24.5 \, \text{J} \)

18.

An object with mass \( m \) moves in one dimension under a force \( F(x) = -kx \), where \( k \) is a constant. The object is displaced by a distance \( x_0 \) from equilibrium and released. What is the total mechanical energy of the system?

a) \( \frac{1}{2} k x_0^2 \) b) \( k x_0 \) c) \( \frac{1}{2} m v_0^2 \) d) \( k x_0^2 \)

19.

A force \( F(x) = -\alpha x^2 \) acts on a particle along the x-axis, where \( \alpha \) is positive. Which of the following expressions correctly gives the potential energy function \( U(x) \) assuming \( U(0) = 0 \)?

a) \( U(x) = \frac{\alpha x^3}{3} \) b) \( U(x) = -\frac{\alpha x^3}{3} \) c) \( U(x) = \alpha x^2 \) d) \( U(x) = -\alpha x^2 \)

20.

A constant power \( P \) is applied to accelerate a car of mass \( m \) from rest on a frictionless road. How does the speed \( v \) of the car depend on time \( t \)?

a) \( v \propto t \) b) \( v \propto \sqrt{t} \) c) \( v \propto t^2 \) d) \( v \) is constant

21.

A 5 kg block is pulled horizontally with a force of 30 N over a rough surface with coefficient of kinetic friction 0.3 for 10 m. What is the net work done on the block?

a) 153 J b) 150 J c) 120 J d) 90 J

22.

An object is moving with speed \( v \) and collides elastically with a stationary object of the same mass. What is the total kinetic energy after the collision?

a) \( \frac{1}{2} m v^2 \) b) \( m v^2 \) c) 0 d) \( \frac{1}{4} m v^2 \)

23.

A force \( F(x) = 3x^2 - 2x + 1 \) (in newtons) acts on an object along the x-axis. Calculate the work done by this force as the object moves from \( x=0 \) to \( x=2 \) m.

a) 6 J b) 8 J c) 10 J d) 12 J

24.

A mass-spring system oscillates with amplitude \( A \) and frequency \( f \). Which of the following expressions represents the maximum power delivered to the system if driven at resonance by an external force?

a) \( P_{max} = \frac{1}{2} k A^2 f \) b) \( P_{max} = k A^2 \pi f \) c) \( P_{max} = b A^2 \omega^2 \) d) \( P_{max} = \frac{1}{2} b A^2 \omega^2 \)

25.

A car of mass 1000 kg accelerates uniformly from rest to 20 m/s in 8 seconds. Calculate the average power output of the engine during this time.

a) 25,000 W b) 30,000 W c) 20,000 W d) 40,000 W

26.

A 2 kg block slides down a frictionless incline of height 5 m. If the block compresses a spring of spring constant 800 N/m at the bottom, what is the maximum compression of the spring?

a) 0.35 m b) 0.5 m c) 0.25 m d) 0.45 m

27.

An object of mass 3 kg is moving at 4 m/s and experiences a resistive force \( F = -kv \) where \( k = 0.6 \, \text{kg/s} \). How much work is done by the resistive force after the object moves 5 m?

a) -12 J b) -30 J c) -24 J d) -40 J

28.

A force \( \vec{F} = (2x \hat{i} + 3y \hat{j}) \, \text{N} \) acts on a particle moving from \( (0,0) \) to \( (3,4) \). What is the total work done by this force along this path?

a) 42 J b) 33 J c) 30 J d) 24 J

29.

A 10 kg block attached to a horizontal spring (spring constant 200 N/m) oscillates with amplitude 0.1 m. What is the total mechanical energy of the system?

a) 1.0 J b) 2.0 J c) 0.5 J d) 1.5 J

30.

A particle moves under a force described by the potential energy function \( U(x) = 4x^3 - 6x^2 + 3 \) J. What is the force \( F(x) \) at \( x=2 \) m?

a) -24 N b) 12 N c) 18 N d) -12 N

Work, Energy, and Power Practice Test

30 Questions – Mixed Conceptual & Calculations