A car can decelerate at -2.47-m/s2 without skidding when coming to rest on a level road. What would its deceleration (in m/s2) be if the road is inclined at 8.7o and the car moves uphill? (Assume the same static friction coefficient.
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To find the deceleration of the car moving uphill on a road inclined at 8.7 degrees, we first need to find the net force acting on the car.
The net force can be calculated using Newton's second law:
F_net = m * a,
where F_net is the net force, m is the mass of the car, and a is the acceleration.
On a level road, the net force is due to friction and given by:
F_friction = m * a_friction,
where a_friction is the deceleration due to friction.
Given that the car can decelerate at -2.47 m/s^2 without skidding on a level road, we know that a_friction = -2.47 m/s^2.
Now, when the road is inclined, the net force includes the force due to gravity:
F_net = F_friction + F_gravity.
The force due to gravity can be calculated using:
F_gravity = m * g * sin(theta),
where g is the acceleration due to gravity (equal to 9.8 m/s^2) and theta is the angle of the incline (8.7 degrees).
Substituting the values into the equation, we have:
F_net = m * a = m * a_friction + m * g * sin(theta).
Rearranging the equation to solve for acceleration:
a = a_friction + g * sin(theta).
Now, plugging in the values:
a = -2.47 m/s^2 + (9.8 m/s^2) * sin(8.7 degrees).
Using a calculator, we can calculate the value of sin(8.7 degrees) which is approximately 0.1504.
a = -2.47 m/s^2 + (9.8 m/s^2) * 0.1504.
Calculating the value, we have:
a = -2.47 m/s^2 + 1.47592 m/s^2.
Adding the values, the deceleration when moving uphill on the inclined road is:
a = -0.99408 m/s^2.
To find the deceleration of the car when moving uphill on an inclined road, we need to consider the components of gravitational force and the force of static friction.
1. Determine the gravitational force component parallel to the incline:
- Gravitational force (Fg) = mass (m) * acceleration due to gravity (g)
- Gravitational force component parallel to the incline (Fg_parallel) = Fg * sin(theta)
(Here, theta is the angle of inclination, which is 8.7 degrees)
2. Determine the maximum static friction force:
- Maximum static friction force (F_static_max) = coefficient of static friction (µ_static) * normal force (N)
(N = m * g, where m is the mass and g is the acceleration due to gravity)
3. Determine the maximum deceleration without skidding:
- Maximum deceleration without skidding (a_max) = F_static_max / m
Now, let's calculate the deceleration of the car when moving uphill on an inclined road.
Given:
- Deceleration without skidding on a level road = -2.47 m/s^2
- Inclination angle (θ) = 8.7 degrees
Step 1:
Fg_parallel = Fg * sin(θ)
= m * g * sin(θ)
Step 2:
N = m * g
F_static_max = µ_static * N
Step 3:
a_max = F_static_max / m
Now, we can substitute the given values and calculate the deceleration:
Fg_parallel = m * g * sin(θ)
= m * 9.8 m/s^2 * sin(8.7 degrees)
N = m * g
F_static_max = µ_static * N
= µ_static * m * g
a_max = F_static_max / m
Finally, we have the deceleration (a_max) when moving uphill on an inclined road.