Principles of Mechanics pp 123-133 | Cite as

# Rolling and Static Equilibrium

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## Abstract

Rolling motion represents the general plane motion of a rigid body. It can be considered as a combination of pure translational motion parallel to a fixed plane plus a pure rotational motion about an axis that is perpendicular to that plane. The axis of rotation usually passes through the center of mass. In Sect. 6.4, we’ve seen that the motion of an object (or a system of particles) can always be considered as a combination of the motion of the object relative to its center of mass plus the motion of its center of mass relative to some origin O.

## 8.1 Rolling Motion

*i*th particle and \(v_{i}'\) is the linear velocity of the ith particle relative to the center of mass. In the case of the general plane motion of a rigid body, the motion can be considered as a combination of pure translational motion of the center of mass plus pure rotational motion about an axis passing through the center of mass and perpendicular to the plane of motion. Therefore, the first term in Eq. 8.1 can be written as

*i*th particle to the center of mass axis. Hence

## 8.2 Rolling Without Slipping

*R*rolling without slipping along the straight track shown in Fig. 8.1. The center of mass of the wheel moves along a straight line, while a point on the rim such as \(\mathrm {P}\) moves in a cycloid path. As the wheel rotates through an angle \(\theta \), its center of mass moves through a distance equal to the arc length

*s*(see Fig. 8.2) given by

*R*from the axis of rotation) and the velocity of a point at the top is \(v_{t}=2R\omega =2v_{cm}\). Note that the angular velocity \(\omega \) of the wheel is the same as its angular velocity if the axis of rotation is at the center of mass.

### Example 8.1

A uniform solid hoop of mass of 32 kg and radius of 1.2 \(\mathrm {m}\) rolls without slipping on a horizontal track where the center of mass speed is 2 \(\mathrm {m}/\mathrm {s}\). Find: (a) the total energy of the hoop and compare it with its total energy if it would slide without rolling; (b) the speed of the hoop at its top and bottom.

### Solution 8.1

### Example 8.2

A uniform solid cylinder, sphere, and hoop roll without slipping from rest at the top of an incline (see Fig. 8.7). Find out which object would reach the bottom first.

### Solution 8.2

### Example 8.3

A marble ball of radius *R* and mass *M* rolls without slipping down the incline shown in Fig. 8.8. Find: (a) its acceleration; (b) the minimum coefficient of static friction that is required to prevent slipping.

### Solution 8.3

### Example 8.4

A string is wrapped around a uniform solid cylinder of radius of *R* and mass of *M* as in Fig. 8.9. If the cylinder is released from rest while the string is fixed in place and assuming that the string does not slip at the cylinder’s surface, find: (a) the acceleration of the center of mass using Newton’s laws (b) the acceleration of the center of mass using energy methods if the cylinder descends a distance \(h(\mathrm {c})\) the tension in the string.

### Solution 8.4

### Example 8.5

A uniform solid sphere of radius *R* and mass *M* is released from rest at the top of an incline at a distance *h* above the ground. If it rolls without slipping, find the speed of the center of mass at the bottom of the incline.

### Solution 8.5

### Example 8.6

A block of mass *m* is attached to a light string that passes over a light pulley and is connected to a uniform solid sphere of radius *R* and mass *M* as in Fig. 8.10. Show that the acceleration of the system is \(a=\displaystyle \frac{g}{1+{7}/5({M_{/m}})}\) when the block is released from rest.

### Solution 8.6

## 8.3 Static Equilibrium

## 8.4 The Center of Gravity

*M*g) and that acts at a single point called the center of gravity Now consider an object that is near the earth’s surface where the force of gravity is assumed to be constant over that range. Equation 8.11 becomes

### Example 8.7

Two blocks of masses \(m_{2}=20\) kg and \(m_{1}=10\) kg are supported by a uniform horizontal beam of length \(L=1.5\mathrm {m}\) and mass \(M=6\) kg (see Fig. 8.14). Find: (a) the normal force exerted by the fulcrum (supporting point) on the beam if it is placed under the center of gravity of the beam; (b) the distance *x* in which \(m_{2}\) must be placed in order for the system to be balanced.

### Solution 8.7

### Example 8.8

A ladder of length *L* and mass \(M=20\) kg rests against a smooth vertical wall as shown in Fig. 8.15. If the center of gravity of the ladder is at a distance of *L*/3 from the base, determine: (a) the minimum coefficient of static friction such that the ladder does not slip; (b) the magnitude and direction of the resultant of the contact forces acting on the ladder at the base; (c) if a man of mass of \(70 \; \mathrm {k}\mathrm {g}\) climbs up the ladder, what is the maximum distance the man can climb before the ladder slips if \(\mu _{s}=0.4.\)

### Solution 8.8

### Example 8.9

A uniform beam of weight *w* and length *L* is held by two supports as in Fig. 8.16. A block of weight \(w_{1}\) is resting on the beam at a distance of *L*/6 from the center of gravity of the beam. Find the magnitude of the forces exerted by the supports on the beam.

### Solution 8.9

### Example 8.10

A man of mass of 80 kg is standing at the end of a uniform beam of mass of 30 kg and length of 12 \(\mathrm {m}\) as shown in Fig. 8.17. Find the tension in the rope and the reaction force exerted by the hinge on the beam.

### Solution 8.10

### Example 8.11

A uniform beam of weight of 120 \(\mathrm {N}\) and length of *L* is in horizontal static equilibrium as in Fig. 8.18. Neglecting the masses of the ropes, find the tension in each string. (The center of mass is at *L*/3 from one end).

### Solution 8.11

### Example 8.12

A solid sphere of mass of 12 kg is in static equilibrium inside the wedge shown in Fig. 8.19. If the surface of the wedge is frictionless, find the forces that the wedge exerts on the sphere.

### Solution 8.12

**Problems**

- 1.
A uniform cylinder of mass 3 kg and radius of 0.05 \(\mathrm {m}\) rolls without slipping along a horizontal surface. Find the total energy of the cylinder at the instant its speed is 2 \(\mathrm {m}/\mathrm {s}.\)

- 2.
A uniform solid cylinder of mass 10 kg and radius of 0.2 \(\mathrm {m}\) rolls up the incline of angle \(45^{\circ }\) with an initial velocity of 15 \(\mathrm {m}/\mathrm {s}\). Find the height in which the cylinder will stop.

- 3.
A wheel of mass 2 kg and radius of 0.05 \(\mathrm {m}\) rolls without slipping with an angular speed of 3 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\) on a horizontal surface. How much work is required to accelerate the wheel to an angular speed of 15 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}.\)

- 4.
A block weighing 1000 \(\mathrm {N}\) is held by a cable that is attached to a uniform rod of weight 500 \(\mathrm {N}\) (see Fig. 8.20). Find (a) the tension in the cable, (b) the horizontal and vertical components of the force exerted on the base of the rod.

- 5.
A uniform sphere of radius

*r*and mass*m*is held by a light string and leans on a frictionless wall as in Fig. 8.21. If the string is attached a distance*d*above the center of the sphere, find (a) the tension in the string, (b) the reaction force exerted by the wall on the sphere. - 6.
Find the minimum force applied at the top of a wheel of mass

*M*and radius*R*to raise it over a step of height*h*as in Fig. 8.22. Assume that the wheel does not slip on the step. - 7.
Three identical uniform blocks each of length

*L*are on top of each other as in Fig. 8.23. Find the maximum value of*h*in order for the stack to be in equilibrium.

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