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Questions and Answers
Newton’s first law states that a particle originally at rest, or moving in a straight line with a constant speed, will remain that way as long as it is not acted upon by an ______ external force.
Newton’s first law states that a particle originally at rest, or moving in a straight line with a constant speed, will remain that way as long as it is not acted upon by an ______ external force.
unbalanced
Newton’s second law states that the acceleration of a mass particle is proportional to the vector resultant force acting on it and is in the direction of this ______.
Newton’s second law states that the acceleration of a mass particle is proportional to the vector resultant force acting on it and is in the direction of this ______.
force
For an object treated as a particle of mass m, the second law can be expressed as dv = ______
For an object treated as a particle of mass m, the second law can be expressed as dv = ______
f
Newton's second law states that the derivative of momentum with respect to time is equal to the ______.
Newton's second law states that the derivative of momentum with respect to time is equal to the ______.
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The term on the left-hand side of the equal sign in equation (3.1.4) is called the ______.
The term on the left-hand side of the equal sign in equation (3.1.4) is called the ______.
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If the work done by the force is independent of the path and depends only on the end points, then the force is called a ______ force.
If the work done by the force is independent of the path and depends only on the end points, then the force is called a ______ force.
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Conservation of mechanical energy states that the change in kinetic energy plus the change in potential energy is ______.
Conservation of mechanical energy states that the change in kinetic energy plus the change in potential energy is ______.
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Newton's law for mass m1 gives m1 ẍ = m1 g − T1 (1)
Newton's law for mass m1 gives m1 ẍ = m1 g − T1 (1)
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T1 = m1 g − m1 ẍ = m1 (g − ẍ)
T1 = m1 g − m1 ẍ = m1 (g − ẍ)
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m2 ÿ = T2 − m2 g (2)
m2 ÿ = T2 − m2 g (2)
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T2 = m2 ÿ + m2 g = m2 ( ÿ + g) = m2 (ẍ + g)
T2 = m2 ÿ + m2 g = m2 ( ÿ + g) = m2 (ẍ + g)
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According to equation (1), the equation L θ̈ = −g sin θ is _________ and not solvable in terms of elementary functions.
According to equation (1), the equation L θ̈ = −g sin θ is _________ and not solvable in terms of elementary functions.
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If θ is small enough and measured in radians, then sin θ ≈ θ. This approximation is valid when θ is _______.
If θ is small enough and measured in radians, then sin θ ≈ θ. This approximation is valid when θ is _______.
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The solution to the equation L θ̈ = −gθ is θ(t) = θ(0) cos ωn t + _______.
The solution to the equation L θ̈ = −gθ is θ(t) = θ(0) cos ωn t + _______.
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The work done by a moment M causing a rotation through an angle θ is given by the equation W = _______.
The work done by a moment M causing a rotation through an angle θ is given by the equation W = _______.
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Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent mass if an external force is applied, and as an equivalent ______ if an external torque is applied.
Equivalent mass and inertia are complementary concepts. A system should be viewed as an equivalent mass if an external force is applied, and as an equivalent ______ if an external torque is applied.
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A pair of spur gears is shown in Figure 3.3.1. The input shaft (shaft 1) is connected to a motor that produces a torque T1 at a speed $\omega1$, and drives the output shaft (shaft 2). One use of such a system is to increase the effective motor ______. The gear ratio N is defined as the ratio of the inp.
A pair of spur gears is shown in Figure 3.3.1. The input shaft (shaft 1) is connected to a motor that produces a torque T1 at a speed $\omega1$, and drives the output shaft (shaft 2). One use of such a system is to increase the effective motor ______. The gear ratio N is defined as the ratio of the inp.
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Some systems composed of translating and rotating parts whose motions are directly coupled can be modeled as a purely translational system or as a purely rotational system, by using the concepts of equivalent mass and equivalent inertia. These models can be derived using ______ energy equivalence.
Some systems composed of translating and rotating parts whose motions are directly coupled can be modeled as a purely translational system or as a purely rotational system, by using the concepts of equivalent mass and equivalent inertia. These models can be derived using ______ energy equivalence.
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Newton’s second law can be used to show that ______ (3.2.1) where ω is the angular velocity of the mass about an axis through a point O fixed in an inertial reference frame and attached to the body (or the body extended), I O is the mass moment of inertia of the body about the point O, and M O is the sum of the moments applied to the body about the point O.
Newton’s second law can be used to show that ______ (3.2.1) where ω is the angular velocity of the mass about an axis through a point O fixed in an inertial reference frame and attached to the body (or the body extended), I O is the mass moment of inertia of the body about the point O, and M O is the sum of the moments applied to the body about the point O.
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The mass moment of inertia I about a specified reference axis is defined as I = r 2 dm (3.2.2) where r is the distance from the reference axis to the mass element dm.
The mass moment of inertia I about a specified reference axis is defined as I = r 2 dm (3.2.2) where r is the distance from the reference axis to the mass element dm.
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If the rotation axis of a homogeneous rigid body does not coincide with the body’s axis of symmetry, but is parallel to it at a distance d, then the mass moment of inertia about the rotation axis is given by the parallel-axis theorem, I = Is + md 2 (3.2.3) where Is is the inertia about the symmetry axis
If the rotation axis of a homogeneous rigid body does not coincide with the body’s axis of symmetry, but is parallel to it at a distance d, then the mass moment of inertia about the rotation axis is given by the parallel-axis theorem, I = Is + md 2 (3.2.3) where Is is the inertia about the symmetry axis
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The pendulum shown in Figure 3.2.4a consists of a concentrated mass m a distance L from point O, attached to a rod of length L. (a) Obtain its equation of motion. (b) Solve the equation assuming that θ is small.
The pendulum shown in Figure 3.2.4a consists of a concentrated mass m a distance L from point O, attached to a rod of length L. (a) Obtain its equation of motion. (b) Solve the equation assuming that θ is small.
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According to equation (1), the equation L θ̈ = −g sin θ is _________ and not solvable in terms of elementary functions.
According to equation (1), the equation L θ̈ = −g sin θ is _________ and not solvable in terms of elementary functions.
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The solution to the equation L θ̈ = −gθ is θ(t) = θ(0) cos ωn t + _______.
The solution to the equation L θ̈ = −gθ is θ(t) = θ(0) cos ωn t + _______.
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The work done by a moment M causing a rotation through an angle θ is given by the equation W = _______.
The work done by a moment M causing a rotation through an angle θ is given by the equation W = _______.
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Newton’s first law states that a particle originally at rest, or moving in a straight line with a constant speed, will remain that way as long as it is not acted upon by an ______ external force.
Newton’s first law states that a particle originally at rest, or moving in a straight line with a constant speed, will remain that way as long as it is not acted upon by an ______ external force.
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According to equation (4.2.1), the equation of motion for the mass-spring system is m ẍ = _______.
According to equation (4.2.1), the equation of motion for the mass-spring system is m ẍ = _______.
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The equilibrium location of the center of mass G is the point marked _______.
The equilibrium location of the center of mass G is the point marked _______.
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The spring force when the mass is displaced a distance x from its equilibrium position is _______.
The spring force when the mass is displaced a distance x from its equilibrium position is _______.
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Elements exerting a resisting force that is a function of velocity are called damping or ______ elements.
Elements exerting a resisting force that is a function of velocity are called damping or ______ elements.
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The most familiar spring is probably the helical coil spring, such as those used in vehicle suspensions and those found in retractable pens. The purpose of the spring in both applications is to provide a ______ force.
The most familiar spring is probably the helical coil spring, such as those used in vehicle suspensions and those found in retractable pens. The purpose of the spring in both applications is to provide a ______ force.
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Sometimes, however, an elastic element is intentionally included in the system, as with a spring in a vehicle suspension. Sometimes the element is not intended to be elastic, but deforms anyway because it is subjected to large forces or torques. This can be the case with the boom or cables of a large crane that lifts a heavy load. In such cases, we must include the deformation and corresponding forces in our analysis.
Sometimes, however, an elastic element is intentionally included in the system, as with a spring in a vehicle suspension. Sometimes the element is not intended to be elastic, but deforms anyway because it is subjected to large forces or torques. This can be the case with the boom or cables of a large crane that lifts a heavy load. In such cases, we must include the deformation and corresponding forces in our analysis.
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According to equation (1), the equation L θ̈ = −g sin θ is _________ and not solvable in terms of elementary functions.
According to equation (1), the equation L θ̈ = −g sin θ is _________ and not solvable in terms of elementary functions.
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If the work done by the force is independent of the path and depends only on the end points, then the force is called a ______ force.
If the work done by the force is independent of the path and depends only on the end points, then the force is called a ______ force.
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The work done by a moment M causing a rotation through an angle θ is given by the equation W = _______.
The work done by a moment M causing a rotation through an angle θ is given by the equation W = _______.
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Newton's law for mass m1 gives m1 ẍ = m1 g − T1 (1)
Newton's law for mass m1 gives m1 ẍ = m1 g − T1 (1)
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According to ______, the restoring force of a spring is given by the equation $f = kx$, where f is the restoring force, x is the compression or extension distance, and k is the spring constant.
According to ______, the restoring force of a spring is given by the equation $f = kx$, where f is the restoring force, x is the compression or extension distance, and k is the spring constant.
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The spring constant, or stiffness, of a spring is denoted by the symbol _____.
The spring constant, or stiffness, of a spring is denoted by the symbol _____.
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The linear force-deflection model states that the restoring force of a spring is directly proportional to the compression or extension distance, and is given by the equation $f = _____$.
The linear force-deflection model states that the restoring force of a spring is directly proportional to the compression or extension distance, and is given by the equation $f = _____$.
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The formula for a coil spring made from round wire is given by the equation $k = ______$, where k is the spring constant, d is the wire diameter, R is the radius of the coil, and n is the number of coils. The shear modulus of elasticity G is a property of the wire material.
The formula for a coil spring made from round wire is given by the equation $k = ______$, where k is the spring constant, d is the wire diameter, R is the radius of the coil, and n is the number of coils. The shear modulus of elasticity G is a property of the wire material.
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An equivalent stiffness k1 can be found from the equation: 1 + 1 + 1 = k1 k k k k, or ______.
An equivalent stiffness k1 can be found from the equation: 1 + 1 + 1 = k1 k k k k, or ______.
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The equivalent stiffness k2 for two springs connected side-by-side can be found from the equation: k2 = k + k = 2k.
The equivalent stiffness k2 for two springs connected side-by-side can be found from the equation: k2 = k + k = 2k.
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The equivalent stiffness ke for the arrangement shown in Figure 4.1.9(b) can be found from the equation: 1/ke = 1/k1 + 1/k2, which gives ______.
The equivalent stiffness ke for the arrangement shown in Figure 4.1.9(b) can be found from the equation: 1/ke = 1/k1 + 1/k2, which gives ______.
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The equivalent stiffness ke for the simplest equivalent arrangement shown in Figure 4.1.9(c) can be found from the equation: 1/ke = 1/k1 + 1/k2, which gives ______.
The equivalent stiffness ke for the simplest equivalent arrangement shown in Figure 4.1.9(c) can be found from the equation: 1/ke = 1/k1 + 1/k2, which gives ______.
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Derive the spring constant expression for a cylindrical rod subjected to an axial force (either tensile or compressive). The rod length is L and its area is A. Solution From mechanics of materials references, for example [Young, 2011], we obtain the forcedeflection relation of a cylindrical rod: L 4L x= f = f EA πED2 where x is the axial deformation of the rod, f is the applied axial force, A is the cross-sectional area, D is the diameter, and E is the modulus of elasticity of the rod material. Rewrite this equation as
Derive the spring constant expression for a cylindrical rod subjected to an axial force (either tensile or compressive). The rod length is L and its area is A. Solution From mechanics of materials references, for example [Young, 2011], we obtain the forcedeflection relation of a cylindrical rod: L 4L x= f = f EA πED2 where x is the axial deformation of the rod, f is the applied axial force, A is the cross-sectional area, D is the diameter, and E is the modulus of elasticity of the rod material. Rewrite this equation as
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The spring constant is the ratio of the applied force f to the resulting deflection x, or
The spring constant is the ratio of the applied force f to the resulting deflection x, or
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Springs for vehicle suspensions are often constructed by strapping together several layers of such springs, as shown in Figure 4.1.4. The value of the total spring constant depends not only on the spring constants of the individual layers, but also on the how they are strapped together, the method of attachment to the axle and chassis, and whether any material to reduce friction has been placed between the layers. ______
Springs for vehicle suspensions are often constructed by strapping together several layers of such springs, as shown in Figure 4.1.4. The value of the total spring constant depends not only on the spring constants of the individual layers, but also on the how they are strapped together, the method of attachment to the axle and chassis, and whether any material to reduce friction has been placed between the layers. ______
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The spring relation for a torsional spring is usually written as
The spring relation for a torsional spring is usually written as
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According to equation (4.2.1), the equation of motion for the mass-spring system is m ẍ = ________.
According to equation (4.2.1), the equation of motion for the mass-spring system is m ẍ = ________.
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The formula for a coil spring made from round wire is given by the equation $k = ________$, where k is the spring constant, d is the wire diameter, R is the radius of the coil, and n is the number of coils. The shear modulus of elasticity G is a property of the wire material.
The formula for a coil spring made from round wire is given by the equation $k = ________$, where k is the spring constant, d is the wire diameter, R is the radius of the coil, and n is the number of coils. The shear modulus of elasticity G is a property of the wire material.
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The equivalent stiffness ke for the simplest equivalent arrangement shown in Figure 4.1.9(c) can be found from the equation: 1/ke = 1/k1 + 1/k2, which gives ________
The equivalent stiffness ke for the simplest equivalent arrangement shown in Figure 4.1.9(c) can be found from the equation: 1/ke = 1/k1 + 1/k2, which gives ________
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The linear force-deflection model states that the restoring force of a spring is directly proportional to the compression or extension distance, and is given by the equation $f = _____$.
The linear force-deflection model states that the restoring force of a spring is directly proportional to the compression or extension distance, and is given by the equation $f = _____$.
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Study Notes
Newton's Laws
- Newton's first law states that a particle originally at rest or moving in a straight line with a constant speed will remain that way as long as it is not acted upon by an external force.
- Newton's second law states that the acceleration of a mass particle is proportional to the vector resultant force acting on it and is in the direction of this force.
- Newton's second law can be expressed as dv = F/m, where dv is the acceleration of the mass particle, F is the vector resultant force, and m is the mass of the particle.
Energy and Work
- The work done by a moment M causing a rotation through an angle θ is given by the equation W = Mθ.
- Conservation of mechanical energy states that the change in kinetic energy plus the change in potential energy is zero.
Rotational Motion
- The equation of motion for rotational motion is L θ̈ = −g sin θ, where L is the length of the pendulum, θ is the angular displacement, and g is the acceleration due to gravity.
- The solution to the equation L θ̈ = −gθ is θ(t) = θ(0) cos ωn t + φ, where ωn is the natural frequency and φ is the phase angle.
Equivalent Mass and Inertia
- A system should be viewed as an equivalent mass if an external force is applied, and as an equivalent inertia if an external torque is applied.
- Equivalent mass and inertia are complementary concepts.
Gears and Gear Ratios
- A pair of spur gears is used to increase the effective motor torque.
- The gear ratio N is defined as the ratio of the input shaft speed to the output shaft speed.
Energy Equivalence
- Some systems composed of translating and rotating parts whose motions are directly coupled can be modeled as a purely translational system or as a purely rotational system, by using the concepts of equivalent mass and equivalent inertia.
- These models can be derived using energy equivalence.
moment of Inertia
- The moment of inertia I about a specified reference axis is defined as I = r^2 dm, where r is the distance from the reference axis to the mass element dm.
- If the rotation axis of a homogeneous rigid body does not coincide with the body's axis of symmetry, but is parallel to it at a distance d, then the mass moment of inertia about the rotation axis is given by the parallel-axis theorem, I = Is + md^2.
Springs and Spring Constants
- The linear force-deflection model states that the restoring force of a spring is directly proportional to the compression or extension distance, and is given by the equation f = kx, where f is the restoring force, x is the compression or extension distance, and k is the spring constant.
- The formula for a coil spring made from round wire is given by the equation k = (πED^4)/(64nR), where k is the spring constant, E is the modulus of elasticity, D is the diameter of the wire, R is the radius of the coil, and n is the number of coils.
- The equivalent stiffness ke for two springs connected in series can be found from the equation: 1/ke = 1/k1 + 1/k2, which gives ke = k1k2/(k1+k2).
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Quiz: Equation of Motion and Tension Forces in a Pulley System Test your understanding of Newton's laws and equation of motion in this quiz. Get ready to solve problems involving pulley systems and calculate the tension forces in the cables. Practice applying energy-based analysis and learn how to derive the equation of motion in terms of displacement. Challenge yourself with example 3.
