A 62.0-kg baseball player slides 3.40 m from third base w/ a speed of 4.35 m/s. If the player comes to rest in third base , (a) how much work was done on the player by friction? (b) what was the coefficient of the kinetic friction between the player and the ground?
(a) The same as his initial kinetic energy
(b) The friction force F is
(Initial kinetic energy)/(sliding distance)
F = (1/2) M V^2/X
Divide that by the player's weight (M g) to get the kinetic friction coefficient.
mu,k = F/(M*g) = V^2/(2 g X)
After plugging in the values, I obtained the following results:
(a) W = K = -587J
(b) muk = 0.285
To find the work done by friction on the baseball player, we can use the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.
(a) The initial kinetic energy of the player can be calculated using the formula:
KE_initial = (1/2) * mass * velocity^2
Substituting the given values:
KE_initial = (1/2) * 62.0 kg * (4.35 m/s)^2
= 492.0245 J
Since the player comes to rest at third base, the final kinetic energy (KE_final) is zero.
The work done by friction can be calculated using the formula:
Work = KE_initial - KE_final
Work = 492.0245 J - 0 J
= 492.0245 J
Therefore, the work done by friction on the player is 492.0245 Joules.
(b) The coefficient of kinetic friction (μ_k) can be calculated using the formula:
Work = frictional force * distance
The work done by friction (492.0245 J) is equal to the product of the frictional force and the distance the player slides (3.40 m). Thus:
492.0245 J = frictional force * 3.40 m
The frictional force can be calculated by dividing the player's weight by the coefficient of kinetic friction:
frictional force = mass * acceleration due to gravity * μ_k
Substituting the given values, the equation becomes:
492.0245 J = (62.0 kg * 9.8 m/s^2) * μ_k * 3.40 m
Simplifying the equation:
μ_k = 492.0245 J / (62.0 kg * 9.8 m/s^2 * 3.40 m)
Finally, calculate μ_k:
μ_k = 0.2272
Therefore, the coefficient of the kinetic friction between the player and the ground is approximately 0.2272.
To find the answer to this question, we need to use the work-energy principle and the equation for work done by friction.
(a) The work done on an object by a force can be calculated using the formula: Work = Force × Distance × cos(θ), where θ is the angle between the force and the displacement.
In this case, the friction force opposes the motion of the player, so the angle between the force and the displacement is 180 degrees (θ = 180°). Therefore, cos(θ) = -1.
The work done by friction can be expressed as: Work = Force × Distance × cos(θ)
However, we need to determine the force of friction first. We can use Newton's second law, which states that the net force acting on an object is equal to the mass times the acceleration (F = m × a).
The player comes to rest, so their final velocity is zero. We can use the equation of motion, v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity (4.35 m/s), a is the acceleration, and s is the displacement. Solving for acceleration:
0^2 = (4.35 m/s)^2 + 2 × a × 3.40 m
-4.35^2 = 6.8a
a = -9.049 m/s^2
Since the acceleration is negative, it means the friction force is in the positive direction opposing the motion.
Using Newton's second law, F = m × a:
Force = mass × acceleration
Force = 62.0 kg × (-9.049 m/s^2)
Force = -560.38 N
Now that we have the force, we can calculate the work done by friction:
Work = Force × Distance × cos(θ)
Work = -560.38 N × 3.40 m × cos(180°)
Work = -560.38 N × 3.40 m × (-1)
Work = 1911.45 J
Therefore, the work done on the player by friction is 1911.45 Joules.
(b) The coefficient of kinetic friction can be found from the formula: Friction Force = coefficient of kinetic friction × Normal Force.
The normal force is the force exerted by the ground on the player, which is equal to the player's weight (mg), where m is the mass and g is the acceleration due to gravity (9.8 m/s^2). Therefore:
Normal Force = (mass of the player) × (acceleration due to gravity)
Normal Force = 62.0 kg × 9.8 m/s^2
Normal Force = 607.6 N
Now we can calculate the coefficient of kinetic friction:
Friction Force = coefficient of kinetic friction × Normal Force
-560.38 N = coefficient of kinetic friction × 607.6 N
Coefficient of kinetic friction = -560.38 N / 607.6 N
Coefficient of kinetic friction = -0.92
Therefore, the coefficient of kinetic friction between the player and the ground is approximately -0.92. Note that the negative sign indicates that frictional force is acting in the opposite direction of motion.