Sally Skier skiis down a rough ice covered 10.0 m high incline into a smooth bowl of radius 4.0 m. Sally starts her descent with a speed of 3.0 m/s at the top of the hill and arrives at the bottom of the bowl (point B) with a speed of 10.3 m/s. Sally has a mass of 65.0 kg.
a. Calculate the work done by friction during Sally’s descent.
b. What is Sally’s apparent weight at the bottom of the bowl?
Sure! Let's break down the problem step by step:
Step 1: Calculate the change in potential energy (ΔPE) as Sally skis down the incline.
The change in potential energy is given by the formula:
ΔPE = m * g * h
where:
m = mass of Sally (65.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the incline (10.0 m)
ΔPE = (65.0 kg) * (9.8 m/s^2) * (10.0 m)
ΔPE = 63700 J
Step 2: Calculate the change in kinetic energy (ΔKE) as Sally skis down the incline and enters the bowl.
The change in kinetic energy is given by the formula:
ΔKE = (1/2) * m * (Vf^2 - Vi^2)
where:
m = mass of Sally (65.0 kg)
Vi = initial velocity (3.0 m/s)
Vf = final velocity (10.3 m/s)
ΔKE = (1/2) * (65.0 kg) * ((10.3 m/s)^2 - (3.0 m/s)^2)
ΔKE = 9168.975 J
Step 3: Calculate the work done by friction (Wf) during Sally's descent.
The work done by friction is given by the formula:
Wf = ΔPE - ΔKE
Wf = 63700 J - 9168.975 J
Wf = 54531.025 J
Therefore, the work done by friction during Sally's descent is 54531.025 J.
Step 4: Calculate Sally's apparent weight at the bottom of the bowl.
The apparent weight of an object is equal to the normal force acting on the object.
In this case, as Sally moves in a vertical circular path, the normal force provides the centripetal force required for circular motion. The net force acting on Sally at the bottom of the bowl is given by:
Net Force = Apparent Weight - Weight - Friction Force
At the bottom of the bowl, the net force acting on Sally must be equal to the centripetal force:
Net Force = Centripetal Force
The centripetal force is given by:
Centripetal Force = m * v^2 / r
where:
m = mass of Sally (65.0 kg)
v = velocity of Sally at the bottom of the bowl (10.3 m/s)
r = radius of the bowl (4.0 m)
Centripetal Force = (65.0 kg) * (10.3 m/s)^2 / 4.0 m
Centripetal Force = 1717.375 N
Since the net force at the bottom of the bowl is equal to the centripetal force, we can write:
Apparent Weight - Weight - Friction Force = Centripetal Force
Solving for the apparent weight:
Apparent Weight = Weight + Friction Force + Centripetal Force
Weight = m * g
Apparent Weight = (65.0 kg) * (9.8 m/s^2) + (54531.025 J / 4.0 m) + 1717.375 N
Apparent Weight = 637.0 N + 13632.75625 N + 1717.375 N
Apparent Weight = 15987.13125 N
Therefore, Sally's apparent weight at the bottom of the bowl is 15987.13125 N.
To find the answers to these questions, we need to use the principles of conservation of energy and the relationship between work, energy, and force.
a. Calculate the work done by friction during Sally’s descent:
To find the work done by friction, we need to calculate the total mechanical energy at the top of the incline and at the bottom of the bowl and then find the difference between them.
The total mechanical energy at the top of the incline consists of kinetic energy and gravitational potential energy:
E1 = KE1 + PE1
KE1 = (1/2) * mass * velocity^2
KE1 = (1/2) * 65.0 kg * (3.0 m/s)^2
KE1 = 292.5 J
PE1 = mass * gravitational acceleration * height
PE1 = 65.0 kg * 9.8 m/s^2 * 10.0 m
PE1 = 6370 J
E1 = 292.5 J + 6370 J
E1 = 6662.5 J
The total mechanical energy at the bottom of the bowl only consists of kinetic energy:
E2 = KE2
KE2 = (1/2) * mass * velocity^2
KE2 = (1/2) * 65.0 kg * (10.3 m/s)^2
KE2 = 3474.095 J
The work done by friction is then the difference between the mechanical energies:
Work_done = E2 - E1
Work_done = 3474.095 J - 6662.5 J
Work_done = -3188.405 J
The work done by friction during Sally's descent is -3188.405 joules (negative sign indicates work done against the direction of motion).
b. What is Sally’s apparent weight at the bottom of the bowl:
To find Sally's apparent weight at the bottom of the bowl, we need to calculate the net force acting on her and then divide it by her mass.
The net force is the centripetal force required to keep Sally moving in a circular path at the bottom of the bowl:
Net_force = mass * centripetal acceleration
Centripetal acceleration = velocity^2 / radius
Centripetal acceleration = (10.3 m/s)^2 / 4.0 m
Centripetal acceleration = 26.765 m/s^2
Net_force = 65.0 kg * 26.765 m/s^2
Net_force = 1741.725 N
Sally's apparent weight is the force of gravity acting on her:
Apparent_weight = mass * gravitational acceleration
Apparent_weight = 65.0 kg * 9.8 m/s^2
Apparent_weight = 637 N
Since the net force acting on Sally at the bottom of the bowl is greater than her weight, her apparent weight is the sum of her weight and the net force:
Apparent_weight = weight + net_force
Apparent_weight = 637 N + 1741.725 N
Apparent_weight = 2378.725 N
Sally's apparent weight at the bottom of the bowl is 2378.725 Newtons.