Systems and Center of Mass: แบบฝึกหัด

A car of mass \(m\) travels around a circular banked curve of radius \(R\). The road is banked at an angle \(\theta\) to the horizontal, and the coefficient of static friction between the tires and the road is \(\mu_s\). What is the maximum constant speed \(v_{max}\) the car can maintain without sliding up the bank?

A smooth wedge of mass \(M\) with an inclination angle \(\theta\) rests on a frictionless horizontal floor. A small block of mass \(m\) is placed on the inclined surface of the wedge. If both the block and the wedge are released from rest, what is the magnitude of the horizontal acceleration of the wedge?

A block of mass \(m\) is placed on a horizontal surface with a coefficient of kinetic friction \(\mu_k\). A constant force \(F\) is applied to the block at an angle \(\theta\) above the horizontal. If the block moves along the surface, what is the magnitude of its horizontal acceleration \(a\)?

A particle of mass \(m\) is initially at rest at position \(x = 0\). It is then acted upon by a position-dependent force \(F(x) = F_0 e^{-ax}\), where \(F_0\) and \(a\) are positive constants. What is the speed of the particle as its position \(x\) approaches infinity?

An object of mass \(m\) is released from rest in a medium that exerts a resistive drag force \(F_d = -kv\), where \(v\) is the instantaneous velocity and \(k\) is a positive constant. How much time \(t\) does it take for the object to reach exactly half of its terminal velocity?

A block of mass \( m \) is held against a vertical wall by a horizontal force \( F \). If the coefficient of static friction between the block and the wall is \( \mu_s \), determine the minimum magnitude of \( F \) required to prevent the block from sliding downwards under gravity \( g \).

A block of mass \( m \) rests on a smooth wedge of mass \( M \) and inclination angle \( \theta \), which in turn rests on a smooth horizontal floor. If the system is released from rest, find the expression for the horizontal acceleration of the wedge as the block slides down.

A system consists of a block of mass \( M \) on a horizontal table with kinetic friction coefficient \( \mu_k \), connected by a light string over a light frictionless pulley to a hanging mass \( m \). If the mass on the table is doubled to \( 2M \), find the expression for the ratio of the new acceleration of the system to its initial acceleration.

A block of mass \( m \) is held against a rough vertical wall by an external force \( F \) that is applied at an angle \( \alpha \) above the horizontal. The coefficient of static friction between the block and the wall is \( \mu_s \).

(a) Determine the minimum force \( F_{min} \) required to prevent the block from sliding down the wall.
(b) Determine the maximum force \( F_{max} \) that can be applied before the block begins to slide upward along the wall.
(c) Under what condition regarding the angle \( \alpha \) and \( \mu_s \) is it impossible to prevent the block from sliding down, regardless of the magnitude of \( F \)?

A small bead of mass \( m \) is constrained to slide without friction along a circular wire hoop of radius \( R \). The hoop is oriented vertically and rotates about its vertical diameter with a constant angular velocity \( \omega \). Let \( \theta \) be the angle measured from the bottom of the hoop to the position of the bead.

(a) Draw a free-body diagram for the bead in a frame rotating with the hoop.
(b) Show that the bead can maintain a stable equilibrium at an angle \( \theta \) such that \( \cos \theta = \frac{g}{R\omega^2} \).
(c) Determine the minimum angular velocity \( \omega_c \) required for the bead to be able to remain in equilibrium at any position other than the bottom of the hoop (\( \theta = 0 \)).