A block is shoved up a 22^∘ slope with an initial speed of 1.4 m / s The coefficient of kinetic

A block is shoved up a 22^∘ slope with an initial speed of...

A block is shoved up a $22^{\circ}$ slope with an initial speed of $1.4 \mathrm{m} / \mathrm{s}$ The coefficient of kinetic friction is $0.70 .$ (a) How far up the slope will the block get? (b) Once stopped, will it slide back down?

Key Concepts

Static Friction

Static friction is the resistive force that prevents relative motion when an object is at rest on a surface. Its maximum value is determined by the product of the coefficient of static friction and the normal force. In problems where an object comes to a stop on an incline, comparing the gravitational component down the slope and the maximum static friction is necessary to decide if the object will remain at rest or begin sliding.

Kinematics on an Incline

Kinematics on an inclined plane involves using the equations of motion for objects experiencing constant acceleration along the plane. This acceleration is influenced by both the gravitational component along the incline and the frictional forces. Analyzing these equations allows one to determine parameters like the distance traveled before coming to a stop, which is central to understanding the dynamics of motion on the incline.

Gravitational Force Decomposition

On an inclined plane, gravity is resolved into two components: one perpendicular to the plane (which influences the normal force) and one parallel to the plane (which drives the motion along the slope). This decomposition is critical for analyzing the motion because it determines both the gravitational pull down the incline and the magnitude of the normal force, which in turn affects frictional forces.

Kinetic Friction

Kinetic friction is the resistive force that acts opposite to the direction of motion when there is sliding contact between surfaces. On an inclined plane, the kinetic frictional force is calculated by multiplying the coefficient of kinetic friction by the normal force. This force is essential in determining how quickly an object decelerates as it moves up the slope.

Transcript

00:01 So we give a block initial speed and the block starts moving up a slope.

00:06 And we know the kinetic coefficient for the slope.

00:11 And we want to know how far the block will get.

00:13 And if the block will slide back down once it gets the maximum value.

00:19 So first we can draw the free body diagram for this situation.

00:23 So we have the normal force following the slope.

00:27 So normal force, force of ground.

00:31 Gravity, fg, and then the friction that is opposing the direction of motion.

00:43 So this is theta, fk, normal force, and gravity.

00:49 In components we have the newton equation as kinetic friction plus mg sine of theta is equal to m a, and n is equal to m g cosine of theta.

01:03 We can combine both of this equation because we know so fk equals mu k n and we can substitute the normal force to get to give the acceleration is equal to g sine of theta plus mu kgg cosine of theta.

01:22 And now we need to solve for the kinematic equation to find the displacement.

01:29 So we solve for v square equals v0 square plus 2a delta x.

01:35 We want the situation where v to define a velocity 0 and we want to find delta x.

01:41 So delta x is equal to v0 squared divided by 2a.

01:51 And this gives us v0 square divided by 2 times g sine of theta plus mu kg full sign of theta.

02:08 And this is 1 .4 meters per second square 2 times 9 .8.

02:16 Times sine of 22 degrees plus 0 .7 full sign of 22 degrees...