A cookie jar is moving up a 40^∘ incline. At a point 55 cm from the bottom of the incline (meas

A cookie jar is moving up a 40^∘ incline. At a point 55 cm...

A cookie jar is moving up a $40^{\circ}$ incline. At a point $55 \mathrm{~cm}$ from the bottom of the incline (measured along the incline), the jar has a speed of $1.4 \mathrm{~m} / \mathrm{s}$. The coefficient of kinetic friction between jar and incline is $0.15 .$ (a) How much farther up the incline will the jar move? (b) How fast will it be going when it has slid back to the bottom of the incline? (c) Do the answers to (a) and (b) increase, decrease, or remain the same if we decrease the coefficient of kinetic friction (but do not change the given speed or location)?

Key Concepts

Effects of Friction Coefficient Variation

This concept deals with how changes in the coefficient of kinetic friction influence motion. A lower coefficient reduces the resistive force, resulting in smaller energy losses due to friction. Consequently, objects may travel farther and retain more speed under the same initial conditions. Understanding this effect is important for predicting how adjustments to friction will alter an object's trajectory and final energy.

Energy Dissipation

Energy dissipation refers to the conversion of mechanical energy into non-mechanical forms, such as heat, due to friction. As an object moves against kinetic friction on an incline, some of its mechanical energy is lost. Accounting for energy dissipation is crucial when applying the conservation of energy, since the energy lost to friction must be considered to accurately predict changes in motion.

Work-Energy Principle

The work-energy principle states that the net work done on an object is equal to its change in kinetic energy. When an object moves along an inclined plane, the work done by gravity and friction converts kinetic energy into potential energy and thermal energy respectively. This principle is key to solving problems involving changes in speed and displacement due to non-conservative forces like friction.

Kinetic Friction

Kinetic friction is the resistive force that opposes the motion of an object sliding over a surface. Its magnitude is proportional to the normal force and the coefficient of kinetic friction. In problems involving motion on an incline, kinetic friction plays a crucial role in energy dissipation and force balance, directly affecting the distance traveled and speed of the object.

Inclined Plane Dynamics

This concept involves the analysis of motion along a slope. It requires resolving forces into components parallel and perpendicular to the incline and understanding how gravity affects the object’s motion along the sloping surface. It is essential for determining the acceleration, net force, and energy changes when objects move on an inclined plane.

Transcript

00:01 So for part a, we know that we're going to use the force analysis in chapter 6, where we know that the normal force is going to be equal to m .g.

00:11 Cosine of theta.

00:12 Where theta, we know to be 40 degrees.

00:15 We also know that the force of kinetic friction is equaling the coefficient of kinetic friction times force normal.

00:23 And we can then say that this is equalling the coefficient of kinetic friction times mg cosine of theta.

00:30 So at this point, we also know that the coefficient of kinetic friction is equal in 0 .15.

00:36 We are going to then use equation 831, and we can then say that the change in thermal energy would be equal to the force friction kinetic multiplied by d.

00:51 So we can say that this is going to be equal to the coefficient of kinetic friction, m .gd, cosine of theta.

00:57 And we know that also using trigonometry, we know that the change in potential energy would be equal to mgd sine of theta.

01:07 We can then use equation 833, and we know that the work done is equaling zero joules.

01:15 We know that the final kinetic energy is equaling zero joules.

01:19 Therefore, we want to find the d, the distance, so we can say that the initial kinetic energy is going to be equal to the change in potential energy plus the change in thermal energy.

01:31 We know that we can then say that one half times mv initial squared would be equal to mgd, and then this would be times sine of theta plus the coefficient of kinetic friction cosine of theta.

01:47 So essentially we have factored out the mgd.

01:51 And we know that the v initial is going to be equal to 1 .4 meters per second.

01:56 Therefore, d, we can say is going to be equal to v.

02:01 Initial square divided by 2g times sine of theta plus the coefficient of kinetic friction mu sub k cosine of theta and this is equal in 0 .13 meters this would be your distance answer for part a for part b we know that here we know that the incline that the object stops on the incline d prime this would be equal to 0 .13 meters plus 0 .55 meters.

02:30 This is equaling 0 .68 meters from the bottom of the incline.

02:39 Given this, we can then use equation 833 again...

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