A box of books is initially at rest a distance D=0.540 m from the end of a wooden board. The co

step 1 So here we have the free butter diagram of the system. We can apply Newton's second law in the X

A box of books is initially at rest a distance D=0.540 m...

A box of books is initially at rest a distance $D=0.540 \mathrm{~m}$ from the end of a wooden board. The coefficient of static friction between the box and the board is $\mu_{s}=0.320$, and the coefficient of kinetic friction is $\mu_{k}=0.250 .$ The angle of the board is increased slowly, until the box just begins to slide; then the board is held at this angle. Find the speed of the box as it reaches the end of the board. -4.55 A block of mass $M_{1}=0.640 \mathrm{~kg}$ is initially at rest on a cart of mass $M_{2}=0.320 \mathrm{~kg}$ with the cart initially at rest on a level air track. The coefficient of static friction between the block and the cart is $\mu_{s}=0.620$, but there is essentially no friction between the air track and the cart. The cart is accelerated by a force of magnitude $F$ parallel to the air track. Find the maximum value of $F$ that allows the block to accelerate with the cart, without sliding on top of the cart.

A box of books is initially at rest a distance $D=0.540 \mathrm{~m}$ from the end of a wooden board. The coefficient of static friction between the box and the board is $\mu_{s}=0.320$, and the coefficient of kinetic friction is $\mu_{k}=0.250 .$ The angle of the board is increased slowly, until the box just begins to slide; then the board is held at this angle. Find the speed of the box as it reaches the end of the board.
-4.55 A block of mass $M_{1}=0.640 \mathrm{~kg}$ is initially at rest on a cart of mass $M_{2}=0.320 \mathrm{~kg}$ with the cart initially at rest on a level air track. The coefficient of static friction between the block and the cart is $\mu_{s}=0.620$, but there is essentially no friction between the air track and the cart. The cart is accelerated by a force of magnitude $F$ parallel to the air track. Find the maximum value of $F$ that allows the block to accelerate with the cart, without sliding on top of the cart.

Key Concepts

Newton's Second Law and Acceleration

Newton's Second Law, stated as F = ma, is fundamental in relating the net forces acting on an object to its resulting acceleration. This law is used to analyze both the accelerating motion on an incline, where gravitational and frictional forces are considered, and the acceleration of a block on a cart where the maximum force must be limited by the friction between the block and the cart to prevent slipping.

Inertial Reference Frames and Relative Motion

In problems involving multiple bodies, such as a block on a cart, it is essential to consider the motion from different reference frames. Understanding inertial frames and relative motion helps in determining the conditions under which the block remains stationary relative to the cart, particularly by ensuring that the acceleration of the block does not exceed the maximum static frictional force available.

Energy Conservation

The principle of energy conservation is applied by equating the initial potential energy of an object to its kinetic energy (and accounting for energy lost due to friction) when determining its speed after moving along a distance. This approach provides a method to calculate the final velocity of an object that has been accelerated by gravity along an inclined plane.

Inclined Plane Dynamics

Inclined plane dynamics involves resolving gravitational forces into components parallel and perpendicular to the plane's surface. This resolution allows for the analysis of motion along the incline, taking into account the gravitational component that drives the motion and the frictional force opposing it, which is critical for calculating acceleration and final velocity on an inclined board.

Kinetic Friction

Kinetic friction acts on an object when it is sliding over a surface, offering resistance proportional to the normal force and characterized by a different coefficient than static friction. In dynamics problems, kinetic friction is used to calculate the deceleration or net force acting on a sliding object, influencing its acceleration and subsequent velocity over a distance.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It determines the maximum force that can be applied before an object starts to move, and is used in problems to establish threshold conditions such as the critical angle at which an object on an incline begins to slide or the maximum force that can be applied to a block without causing relative motion.

Transcript

00:01 So here we have the free butter diagram of the system.

00:04 We can apply newton's second law in the x direction.

00:09 This would be equal to the mass times the acceleration in the x direction.

00:12 We have translational equilibrium in the x direction, so this will be equal to zero.

00:17 And we can say that then m .g.

00:20 Sign of theta minus the maximum force of static friction would be equal to zero.

00:30 Or we can say that then the maximum magnitude of the static frictional force equaling m g sine of theta.

00:42 This would then be equal to the coefficient of static friction multiplied by the normal force.

00:47 We can see that in the wire direction we have translational equilibrium as well.

00:55 So this would be equal to zero.

00:57 And we have then n n minus m g cosine of theta equal.

01:07 Zero or of course and equalling m g cosine of theta so from this you can plug this into here and we see that then mg sign of theta would be equal to the coefficient of static friction mg cosine of theta m cancels out g cancels out and we have then that the coefficient of static friction equals tangent of theta.

01:51 And so theta would be equal to arc tan of the coefficient of static friction.

01:58 0 .320 and this is giving us then 17 .74 degrees.

02:08 Now we're going to then calculate the net force on the blocks after it starts sliding.

02:15 So after it starts sliding, we're we can apply newton's second law in the x direction.

02:23 Now we don't have translational equilibrium.

02:26 So here we have m .g...

Please give Ace some feedback