During a rockslide, a 520 kg rock slides from rest down a hillside that is 500 m long and 300

step 1 Well, we have mass of rock m is equal to 520 kilograms. It slides from rest and friction coeffic

During a rockslide, a 520 kg rock slides from rest down a...

During a rockslide, a $520 \mathrm{~kg}$ rock slides from rest down a hillside that is $500 \mathrm{~m}$ long and $300 \mathrm{~m}$ high. The coefficient of kinetic friction between the rock and the hill surface is $0.25 .$ (a) If the gravitational potential energy $U$ of the rock-Earth system is zero at the bottom of the hill, what is the value of $U$ just before the slide? (b) How much energy is transferred to thermal energy during the slide? (c) What is the kinetic energy of the rock as it reaches the bottom of the hill? (d) What is its speed then?

During a rockslide, a $520 \mathrm{~kg}$ rock slides from rest down a hillside that is $500 \mathrm{~m}$ long and $300 \mathrm{~m}$ high. The coefficient of kinetic friction between the rock and the hill surface is $0.25 .$ (a) If the gravitational potential energy $U$ of the rock-Earth system is zero at the bottom of the hill, what is the value of $U$ just before the slide?
(b) How much energy is transferred to thermal energy during the slide? (c) What is the kinetic energy of the rock as it reaches the bottom of the hill? (d) What is its speed then?

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is defined by the equation U = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above a reference level. This concept is crucial for analyzing systems where an object's elevation leads to stored energy that can be converted into kinetic energy during motion.

Kinetic Friction and Thermal Energy

Kinetic friction is the resistive force that acts against the motion of an object sliding over a surface. The work done by kinetic friction is converted into thermal energy, which represents the loss of mechanical energy from the system. Understanding this concept is fundamental when analyzing energy dissipation and non-conservative forces in dynamics.

Conservation of Energy

The conservation of energy principle states that energy within a closed system remains constant, although it can change forms. In systems involving friction, the initial energy (usually gravitational potential energy) is divided into kinetic energy and thermal energy due to energy losses from non-conservative forces. This concept is essential for solving problems that involve energy transfer and transformation.

Kinetic Energy

Kinetic energy is the energy of motion, given by the equation K = 1/2 mv^2, where m is the mass of the object and v is its speed. It quantifies the energy that an object possesses as a result of its motion. This concept is key in analyzing the final state of objects that have undergone acceleration due to the conversion of potential energy into kinetic energy.

Transcript

00:01 Well, we have mass of rock m is equal to 520 kilograms.

00:14 It slides from rest and friction coefficient f is equal to 0 .25.

00:27 This is the coefficient of friction.

00:34 Is 500 meters long so d is equal to 500 meters and height of hill h is equal to 300 meters and height of hill h is equal to 300 meters and the gravitational potential energy of the rock after the slide u b is equal to zero now let's solve part a the change in gravitational potential energy is delta u is equal to m, g h, and delta u is equal to 520 multiplied by 9 .8, multiply by 300 and this is equal to u -t minus ub, ub and this is equal to 1 .53 multiplied by 10 to the power 6 joules.

02:02 And ut is the potential energy before the slide, a top of the hill.

02:09 And ut is equal to 1 .53 multiply by 10 to the power 6 joules.

02:20 All right.

02:23 Now, let's solve part b.

02:32 Well, take the gravitational force direction to be the negative wire.

02:38 And perpendicular to be the x -axis and the applied force in y direction are the weight w and y component of the normal force which is r y so r y and this is part b so r y the y the y the y component is equal to minus w and which is equal to r cosine theta and this can be written is minus m g and this is equal to 520 multiply by 9 .8 and therefore we have well, here, ry is 520 multiplied with 9 .8 since minus minus is plus, where theta, the angle between the normal force and the positive by direction.

04:03 So theta can be estimated by the hill dimension.

04:07 So sine theta is equal to 300 divided by 500.

04:19 It's in fact h divided by d okay so theta is equal to theta is equal to 36 .9 degrees 36 .9 degrees and therefore our cosine of 36 .9 degrees is equal to 5 ,096 .96 newtons 5 ,000 906 newtons and therefore r is equal to 407 5 .2 okay and the change in thermal energy delta e is equal to the magnitude of the work from the friction kinetic force so delta e thermal is equal to r f cosine phi d where phi is the angle between the kinetic friction force and the direction of motion and therefore we have delta delta d thermal is equal to 2 multiply by 0 .25 multiply by cosine of 180 5 is equal to 180 multiplied by d is 500 and this is equal to minus 50 9 3 double 9 9 jules 9 3 9 9 9 9 9 jules all right so it's equal to 5 .093 2 .9 jules.

06:50 And the negative sign indicates the decrease in the potential energy.

06:59 Okay.

07:00 Now let's solve part c...

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