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18 problems tagged with rigid body balance
P0648
Intermediate Mechanics › Rotational MotionStatic Equilibrium of a Cube on an Incline
A uniform wooden cube of mass $m$ and side length $l$ is placed on a plane inclined at an angle $\theta$.
- If the block remains at rest, find the magnitude of the normal force and the position of its line of action.
- An upward pushing force $F$, parallel to the incline and passing through the cube's center, is applied. The cube remains at rest. The coefficient of static friction is $\mu$. What are the new magnitude and line of action of the normal force? Are there constraints on the magnitude of $F$?
P0649
Intermediate Mechanics › Rotational MotionStatic Friction Condition for Gripping Tongs
In Figure 5.34, AOB is a pair of equal-arm tongs, with a frictionless pivot at O. A force is applied at A and B to grip a cylindrical object C. The weight of the cylindrical object is to be ignored.
- What factors determine whether the tongs can grip the cylinder?
- If the setup is just able to grip cylinder C (i.e., it is on the verge of slipping out), what condition must these factors satisfy?
P0650
Advanced Mechanics › Rotational MotionTension in a Hanging Uniform Chain
As shown in Figure, a uniform chain of weight G is hung on two hooks of equal height. The ends of the chain make an angle $\theta$ with the horizontal.
- Find the force exerted on the ends of the chain.
- Find the tension at the lowest point of the chain.
P0651
Advanced Mechanics › Rotational MotionSliding vs Tipping of a Block on an Incline
A uniform rectangular block with base length a and height b is placed on an inclined plane. The coefficient of static friction between the plane and the block is $\mu$. The inclination angle of the plane, $\theta$, is gradually increased from zero. At some critical angle $\theta_0$, the block will either start to slide or to tip over.
- Find the critical angle $\theta_0$ for the block to start sliding.
- Find the critical angle $\theta_0$ for the block to start tipping over.
- State the conditions that determine whether the block will slide first or tip over first.
P0652
Advanced Mechanics › Rotational MotionEquilibrium of Two Hinged Rods
As shown in Figure 5.32, a thin rod AB is connected to the ceiling by a hinge at A. Its lower end is hinged to another thin rod BC. The two rods have equal length, and their motion is restricted to the vertical plane shown. Friction at the hinges is negligible. An appropriate external force $\vec{F}$ is applied at C to keep the rods in equilibrium in the position shown, where the angle between the rods is $90^\circ$ and C is directly below A.
- Regardless of the masses of the two rods, what is the possible range of directions for the external force $\vec{F}$?
- If the mass of rod AB is $m_1 = 1.0$ kg and the mass of rod BC is $m_2 = 2.0$ kg, find the magnitude and direction of the external force $\vec{F}$.
P0653
Advanced Mechanics › Rotational MotionStability of a Suspended Rod in Water
As shown in Figure, a uniform thin wooden rod of length $l$ and density $\rho$ is suspended vertically by a thin string from its top end. A bucket of water (density $\rho_0$) is slowly lifted, gradually immersing the rod. When the length of the rod immersed in water exceeds a critical length $l'$, the rod becomes unstable and starts to tilt. Find $l'$.
P0654
Advanced Mechanics › Rotational MotionMaximum Force on a Rod Before Slipping
A rod of mass $m=50$ kg and length $L$ stands on a horizontal surface with a coefficient of maximum static friction $\mu=0.3$. The top of the rod is held by a rope fixed to the ground, making an angle $\theta=30^\circ$ with the rod, as shown in Figure. A horizontal force $F$ is applied to the rod at a height $h$ from the ground.
- If $h=2L/5$, what is the maximum value of $F$ before the rod slips?
- What is the situation if the point of application is moved to $h=4L/5$?
- For what values of $h$ will the rod never slip, regardless of the magnitude of $F$?
P0655
Advanced Mechanics › Rotational MotionEquilibrium of Sphere and Cube on Incline
As shown in Figure, a uniform sphere A of weight $G_A = 10$ N and a uniform cubic block B of weight $G_B = 20$ N are placed on a fixed inclined plane with an angle $\alpha=30^\circ$. The sphere's diameter and the cube's side length are both $2R=0.2$ m. A horizontal force $P=15$ N acts on block B, parallel to the base of the incline. The system is in static equilibrium.
- Find the magnitude and direction of the friction forces exerted by the inclined plane on A and B.
- Find the distance between the line of action of the resultant normal force $N_B$ on B and the center of mass O' of B.
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