90 Questions
Which type of force does not satisfy properties 1 and 2 and is path-dependent?
Nonconservative force
What is the definition of the mechanical energy of a system?
The sum of the kinetic and potential energies of a system
What happens to the mechanical energy of a system when nonconservative forces act within the system?
It decreases
What type of force transforms the mechanical energy of a system into internal energy?
Friction force
What is the relationship between conservative forces and potential energy?
The work done by a conservative force equals the change in potential energy of the system
What is the equation for the internal work done by a conservative force as a particle moves along the x axis?
$W_{int} = \int F_x dx$
When is the potential energy difference negative according to Equation 7.27?
When F_x and dx are in the same direction
Which of the following is a property of conservative forces?
The work done by a conservative force on a particle moving through a closed path is always zero.
Which of the following is an example of a conservative force?
Gravitational force
What is the equation for the work done by the gravitational force on an object moving between two points near the Earth's surface?
$W_g = mgy_i - mgy_f$
What is the equation for the work done by the spring force on an object in an object-spring system?
$W_s = 2kx_i - 2kx_f$
What is the equation for the work done by a conservative force on an object as the system changes from one configuration to another?
$W_{int} = U_i - U_f$
What does the subscript 'int' in Equation 7.24 represent?
Internal energy
What is the potential energy associated with a conservative force?
Elastic potential energy
According to Equation 7.29, what is the relationship between the conservative force and the potential energy function?
The conservative force is the negative derivative of the potential energy function
What does the slope of a graph of U(x) versus x represent?
The negative of the x component of the force on the object
In the block-spring system, where is the position of stable equilibrium?
x = 0
What is the potential energy function for the block-spring system?
U(x) = 1/2kx^2
What is the relationship between the force exerted by the spring and the slope of the U-versus-x curve in the block-spring system?
The force is the negative of the slope
In the ball rolling in a bowl system, where is the position of stable equilibrium?
x = 0
What is the relationship between the conservative force and the U-versus-x curve in the particle moving along the x axis system?
The force is the negative of the U-versus-x curve
Which of the following statements is true about the particle's motion when the force is positive and the particle is displaced to the right?
The particle accelerates away from the equilibrium position.
When the particle is at x = 0 and is displaced to the left, what is the sign of the force?
Negative
What type of equilibrium does the position x = 0 represent in this situation?
Unstable equilibrium
Which of the following examples represents a configuration of a system in unstable equilibrium?
A pencil balanced on its point
What type of equilibrium corresponds to configurations where U(x) for the system is a maximum?
Unstable equilibrium
What happens to a ball lying on a flat, horizontal surface when it is slightly displaced from its position of neutral equilibrium?
It experiences no change in motion.
When the force is positive and the particle is displaced to the right, what is the sign of the slope of U(x)?
Positive
When the force is negative and the particle is displaced to the left, what is the sign of the slope of U(x)?
Positive
What type of equilibrium does the position x = 0 represent for the particle?
Unstable equilibrium
True or false: The work done by a conservative force on a particle moving between any two points is dependent on the path taken by the particle.
False
True or false: The work done by a conservative force on a particle moving through any closed path is zero.
True
True or false: The gravitational force is an example of a conservative force.
True
True or false: The work done by the gravitational force on an object moving between any two points near the Earth's surface can be expressed as $W_g = 2mg(y_f - y_i)$.
True
True or false: The spring force is conservative because the work done by it depends only on the initial and final y coordinates of the object.
False
True or false: A potential energy can be associated with a force acting between members of a system if the force is conservative.
True
True or false: The work done by a conservative force on an object as the system changes from one configuration to another is given by the equation $W_{int} = U_i - U_f$.
True
True or false: Work done by an external agent on a system is internal to the system.
False
True or false: Nonconservative forces satisfy properties 1 and 2.
False
True or false: The mechanical energy of a book-surface system remains constant when nonconservative forces act within the system.
False
True or false: The work done on a member of a system by a conservative force depends on the path taken by the moving member.
False
True or false: The work done against the force of kinetic friction depends on the path taken.
True
True or false: The work done by a conservative force within a system equals the negative change in the potential energy of the system.
True
True or false: The potential energy of a system can be defined with respect to a reference configuration.
True
True or false: The value of Ui is often taken to be zero for the reference configuration.
True
True or false: The x component of a conservative force acting on a member within a system equals the negative derivative of the potential energy of the system with respect to x.
True
True or false: The slope of a graph of U(x) versus x represents the magnitude of the force on the object.
False
True or false: The x = 0 position for a block-spring system is one of unstable equilibrium.
False
True or false: The x = 0 position for a particle moving along the x axis under the influence of a conservative force is one of stable equilibrium.
True
True or false: The potential energy of a block-spring system is given by Us = 1/2 kx^2.
True
True or false: The potential energy function for a ball rolling about in the bottom of a bowl is U(x) = 1/2 kx^2.
False
The particle accelerates away from the equilibrium position when the force is positive and the particle is displaced to the right.
True
The force is negative when the particle is displaced to the left because the slope of the potential energy function is positive.
True
The position x=0 represents a position of unstable equilibrium because any displacement from this point results in a force that pushes the particle farther away from equilibrium.
True
A pencil balanced on its point is an example of a system in stable equilibrium.
False
Configurations of a system in unstable equilibrium correspond to those for which the potential energy function U(x) is a maximum.
True
Neutral equilibrium occurs when the potential energy function U(x) is constant over some region.
True
Small displacements of an object from a position of neutral equilibrium produce restoring forces.
False
A ball lying on a flat, horizontal surface is an example of an object in stable equilibrium.
False
For a system in unstable equilibrium, the force exerted by the spring is always positive.
False
Which two properties define a conservative force?
The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle, and the work done by a conservative force on a particle moving through any closed path is zero.
What is an example of a conservative force?
The gravitational force and the force exerted by an ideal spring are examples of conservative forces.
What is the equation for the work done by the gravitational force on an object moving between two points near the Earth's surface?
$W_g = 2mg(y_f - y_i)$
What is the equation for the work done by the spring force in an object-spring system?
$W_s = \frac{1}{2} kx_i^2 - \frac{1}{2} kx_f^2$
What is the equation for the internal work done by a conservative force as a system changes from one configuration to another?
$W_{int} = U_i - U_f$
What is the potential energy function for a block-spring system?
$U_s = \frac{1}{2} kx^2$
True or false: The work done by a conservative force within a system equals the negative change in the potential energy of the system.
True
What is the meaning of unstable equilibrium in this context?
Unstable equilibrium is a configuration where any displacement from the equilibrium position results in a force that pushes the particle farther away from equilibrium.
What is an example of a configuration in unstable equilibrium?
A pencil balanced on its point is an example of a configuration in unstable equilibrium.
What is the meaning of neutral equilibrium in this context?
Neutral equilibrium is a configuration where small displacements of an object from a position in this region produce neither restoring nor disrupting forces.
What is an example of an object in neutral equilibrium?
A ball lying on a flat, horizontal surface is an example of an object in neutral equilibrium.
What type of equilibrium does the position x = 0 represent in this situation?
The position x = 0 represents a position of unstable equilibrium.
What happens to a ball lying on a flat, horizontal surface when it is slightly displaced from its position of neutral equilibrium?
When a ball lying on a flat, horizontal surface is slightly displaced from its position of neutral equilibrium, it will experience restoring forces that attempt to bring it back to the equilibrium position.
What is the relationship between conservative forces and potential energy?
Conservative forces are associated with potential energy. The work done by a conservative force within a system equals the negative change in potential energy of the system.
What type of force transforms the mechanical energy of a system into internal energy?
Nonconservative forces transform the mechanical energy of a system into internal energy.
What is the equation for the work done by a conservative force on an object as the system changes from one configuration to another?
The equation for the work done by a conservative force is given by $W = -\Delta U$, where $W$ is the work done and $\Delta U$ is the change in potential energy.
What relationship exists between the conservative force and the potential energy function? Explain in terms of Equation 7.29.
According to Equation 7.29, the x component of a conservative force acting on a member within a system equals the negative derivative of the potential energy of the system with respect to x. Therefore, the conservative force is related to the potential energy function through this equation.
What does the slope of a graph of U(x) versus x represent?
The slope of a graph of U(x) versus x represents the negative of the magnitude of the force on the object.
Explain why the x = 0 position for a block-spring system is one of stable equilibrium.
The x = 0 position for a block-spring system is one of stable equilibrium because any movement away from this position results in a force directed back toward x = 0. In general, configurations of a system in stable equilibrium correspond to those for which U(x) for the system is a minimum.
What is the relationship between the force exerted by the spring and the slope of the U(x) curve in the block-spring system?
The force exerted by the spring is equal to 2kx, which corresponds to the negative slope of the U(x) curve in the block-spring system.
Explain why the x = 0 position for a particle moving along the x axis under the influence of a conservative force is one of stable equilibrium.
The x = 0 position for a particle moving along the x axis under the influence of a conservative force is one of stable equilibrium because any movement away from this position results in a force directed back toward x = 0. In general, configurations of a system in stable equilibrium correspond to those for which U(x) for the system is a minimum.
What is the equation for the work done by a conservative force on an object as the system changes from one configuration to another?
The equation for the work done by a conservative force on an object as the system changes from one configuration to another is given by: $W_{\text{int}} = U_i - U_f$.
What is the potential energy of a block-spring system?
The potential energy of a block-spring system is given by: $U_s = \frac{1}{2} kx^2$.
What is the definition of a nonconservative force?
A force is nonconservative if it does not satisfy properties 1 and 2 above. The work done by a nonconservative force is path-dependent.
What is the equation for the mechanical energy of a system?
$E_{mech} = K + U$
What happens to the mechanical energy of a system when nonconservative forces act within the system?
Nonconservative forces acting within a system cause a change in the mechanical energy of the system.
What is the equation for the work done by a nonconservative force?
$W_{int} = \Delta E_{mech}$
What is the equation for the work done by a conservative force on a member of a system?
$W_{int} = -\Delta U$
What is the equation for the potential energy function for a system?
$U_f - U_i = -W_{int}$
What is the relationship between the work done by a conservative force and the change in potential energy?
The work done within the system by a conservative force equals the negative of the change in the potential energy of the system.
Quiz: Understanding the Significance of U i and U f in Potential Energy Learn about the importance of assigning values to U i in the reference configuration and how it affects U f (x). Discover why any nonzero value of U i only results in a constant shift in U f (x) and explore the physical meaningfulness of the change in potential energy.
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