A grocery cart is being pushed with a force of 450 N at an angle of 30.0° to the horizontal. If the mass of the cart
and the groceries is 42 kg, (a) Calculate the force of friction if the coefficient of friction is 0.60. (b) Determine the
acceleration of the cart
To calculate the force of friction and the acceleration of the cart, we need to break down the forces acting on the cart.
Let's start with calculating the force of friction.
(a) Force of friction:
The force of friction can be calculated using the formula:
Force of friction = coefficient of friction * normal force
The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the weight of the cart and groceries.
Normal force = mass * gravity
where mass = 42 kg (given) and gravity = 9.8 m/s^2 (acceleration due to gravity).
Normal force = 42 kg * 9.8 m/s^2 = 411.6 N
Now, substitute the values into the formula to find the force of friction:
Force of friction = 0.60 * 411.6 N = 246.96 N
Therefore, the force of friction acting on the cart is 246.96 N.
(b) Acceleration of the cart:
To calculate the acceleration of the cart, we need to consider the net force acting on it. The net force is the difference between the applied force (450 N at an angle of 30.0° to the horizontal) and the force of friction.
Net force = applied force - force of friction
The applied force can be broken down into horizontal and vertical components:
Horizontal component = applied force * cos(angle)
Vertical component = applied force * sin(angle)
Horizontal component = 450 N * cos(30.0°) = 450 N * 0.866 = 389.7 N
Vertical component = 450 N * sin(30.0°) = 450 N * 0.5 = 225 N
Now, calculate the net force:
Net force = horizontal component - force of friction
Net force = 389.7 N - 246.96 N = 142.74 N
Finally, we can use Newton's second law of motion to calculate the acceleration:
acceleration = net force / mass
acceleration = 142.74 N / 42 kg = 3.399 m/s^2
Therefore, the acceleration of the cart is approximately 3.399 m/s^2.